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id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko" class=""><span>기부</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%ED%95%A8%EC%88%98" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다" class=""><span>계정 만들기</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%ED%95%A8%EC%88%98" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%ED%95%A8%EC%88%98" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%ED%95%A8%EC%88%98" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <ul id="toc-정의-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-예" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#예"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>예</span> </div> </a> <ul id="toc-예-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-종류" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#종류"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>종류</span> </div> </a> <button aria-controls="toc-종류-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>종류 하위섹션 토글하기</span> </button> <ul id="toc-종류-sublist" class="vector-toc-list"> <li id="toc-단사_함수와_전사_함수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#단사_함수와_전사_함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>단사 함수와 전사 함수</span> </div> </a> <ul id="toc-단사_함수와_전사_함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-특별한_정의역·공역을_갖는_함수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#특별한_정의역·공역을_갖는_함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>특별한 정의역·공역을 갖는 함수</span> </div> </a> <ul id="toc-특별한_정의역·공역을_갖는_함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-조각마다_정의된_함수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#조각마다_정의된_함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>조각마다 정의된 함수</span> </div> </a> <ul id="toc-조각마다_정의된_함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-다가_함수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#다가_함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>다가 함수</span> </div> </a> <ul id="toc-다가_함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-부분_정의_함수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#부분_정의_함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>부분 정의 함수</span> </div> </a> <ul id="toc-부분_정의_함수-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-연산" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#연산"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>연산</span> </div> </a> <button aria-controls="toc-연산-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>연산 하위섹션 토글하기</span> </button> <ul id="toc-연산-sublist" class="vector-toc-list"> <li id="toc-상과_원상" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#상과_원상"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>상과 원상</span> </div> </a> <ul id="toc-상과_원상-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-역함수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#역함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>역함수</span> </div> </a> <ul id="toc-역함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-합성" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#합성"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>합성</span> </div> </a> <ul id="toc-합성-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-제한" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#제한"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>제한</span> </div> </a> <ul id="toc-제한-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-호환되는_함수족으로_유도되는_함수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#호환되는_함수족으로_유도되는_함수"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>호환되는 함수족으로 유도되는 함수</span> </div> </a> <ul id="toc-호환되는_함수족으로_유도되는_함수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-점별_연산" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#점별_연산"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>점별 연산</span> </div> </a> <ul id="toc-점별_연산-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-역사" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역사"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>역사</span> </div> </a> <button aria-controls="toc-역사-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>역사 하위섹션 토글하기</span> </button> <ul id="toc-역사-sublist" class="vector-toc-list"> <li id="toc-어원" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#어원"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>어원</span> </div> </a> <ul id="toc-어원-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="목차" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="목차 토글" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">목차 토글</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">함수</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="다른 언어로 문서를 방문합니다. 120개 언어로 읽을 수 있습니다" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-120" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">120개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Funksie" title="Funksie – 아프리칸스어" lang="af" hreflang="af" data-title="Funksie" data-language-autonym="Afrikaans" data-language-local-name="아프리칸스어" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik) – 독일어(스위스)" lang="gsw" hreflang="gsw" data-title="Funktion (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="독일어(스위스)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%88%B5%E1%88%A8%E1%8A%AB%E1%89%A2" title="አስረካቢ – 암하라어" lang="am" hreflang="am" data-title="አስረካቢ" data-language-autonym="አማርኛ" data-language-local-name="암하라어" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Funci%C3%B3n_matematica" title="Función matematica – 아라곤어" lang="an" hreflang="an" data-title="Función matematica" data-language-autonym="Aragonés" data-language-local-name="아라곤어" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9" title="دالة – 아랍어" lang="ar" hreflang="ar" data-title="دالة" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AF%D8%A7%D9%84%D8%A9" title="دالة – 모로코 아랍어" lang="ary" hreflang="ary" data-title="دالة" data-language-autonym="الدارجة" data-language-local-name="모로코 아랍어" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica" title="Función matemática – 아스투리아어" lang="ast" hreflang="ast" data-title="Función matemática" data-language-autonym="Asturianu" data-language-local-name="아스투리아어" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Funksiya_(riyaziyyat)" title="Funksiya (riyaziyyat) – 아제르바이잔어" lang="az" hreflang="az" data-title="Funksiya (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="아제르바이잔어" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – 바슈키르어" lang="ba" hreflang="ba" data-title="Функция (математика)" data-language-autonym="Башҡортса" data-language-local-name="바슈키르어" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Funkc%C4%97j%C4%97" title="Funkcėjė – Samogitian" lang="sgs" hreflang="sgs" data-title="Funkcėjė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Функцыя (матэматыка) – 벨라루스어" lang="be" hreflang="be" data-title="Функцыя (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Функцыя (матэматыка) – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Функцыя (матэматыка)" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Функция – 불가리아어" lang="bg" hreflang="bg" data-title="Функция" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%AB%E0%A4%82%E0%A4%95%E0%A5%8D%E0%A4%B6%E0%A4%A8_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="फंक्शन (गणित) – Bhojpuri" lang="bh" hreflang="bh" data-title="फंक्शन (गणित)" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AA%E0%A7%87%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="অপেক্ষক (গণিত) – 벵골어" lang="bn" hreflang="bn" data-title="অপেক্ষক (গণিত)" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – 보스니아어" lang="bs" hreflang="bs" data-title="Funkcija (matematika)" data-language-autonym="Bosanski" data-language-local-name="보스니아어" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Funci%C3%B3" title="Funció – 카탈로니아어" lang="ca" hreflang="ca" data-title="Funció" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%D9%86%DA%A9%D8%B4%D9%86_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="فانکشن (ماتماتیک) – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="فانکشن (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Funkce_(matematika)" title="Funkce (matematika) – 체코어" lang="cs" hreflang="cs" data-title="Funkce (matematika)" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функци (математика) – 추바시어" lang="cv" hreflang="cv" data-title="Функци (математика)" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffwythiant" title="Ffwythiant – 웨일스어" lang="cy" hreflang="cy" data-title="Ffwythiant" data-language-autonym="Cymraeg" data-language-local-name="웨일스어" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Funktion_(matematik)" title="Funktion (matematik) – 덴마크어" lang="da" hreflang="da" data-title="Funktion (matematik)" data-language-autonym="Dansk" data-language-local-name="덴마크어" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik) – 독일어" lang="de" hreflang="de" data-title="Funktion (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Συνάρτηση – 그리스어" lang="el" hreflang="el" data-title="Συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics) – 영어" lang="en" hreflang="en" data-title="Function (mathematics)" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Funkcio_(matematiko)" title="Funkcio (matematiko) – 에스페란토어" lang="eo" hreflang="eo" data-title="Funkcio (matematiko)" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_(matem%C3%A1tica)" title="Función (matemática) – 스페인어" lang="es" hreflang="es" data-title="Función (matemática)" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Funktsioon_(matemaatika)" title="Funktsioon (matemaatika) – 에스토니아어" lang="et" hreflang="et" data-title="Funktsioon (matemaatika)" data-language-autonym="Eesti" data-language-local-name="에스토니아어" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_(matematika)" title="Funtzio (matematika) – 바스크어" lang="eu" hreflang="eu" data-title="Funtzio (matematika)" data-language-autonym="Euskara" data-language-local-name="바스크어" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9" title="تابع – 페르시아어" lang="fa" hreflang="fa" data-title="تابع" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Funktio" title="Funktio – 핀란드어" lang="fi" hreflang="fi" data-title="Funktio" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Cakacaka_(fika)" title="Cakacaka (fika) – 피지어" lang="fj" hreflang="fj" data-title="Cakacaka (fika)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="피지어" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Funksj%C3%B3n" title="Funksjón – 페로어" lang="fo" hreflang="fo" data-title="Funksjón" data-language-autonym="Føroyskt" data-language-local-name="페로어" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fonction_(math%C3%A9matiques)" title="Fonction (mathématiques) – 프랑스어" lang="fr" hreflang="fr" data-title="Fonction (mathématiques)" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Funksion" title="Funksion – 북부 프리지아어" lang="frr" hreflang="frr" data-title="Funksion" data-language-autonym="Nordfriisk" data-language-local-name="북부 프리지아어" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Feidhm_(matamaitic)" title="Feidhm (matamaitic) – 아일랜드어" lang="ga" hreflang="ga" data-title="Feidhm (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="아일랜드어" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%87%BD%E6%95%B8" title="函數 – 간어" lang="gan" hreflang="gan" data-title="函數" data-language-autonym="贛語" data-language-local-name="간어" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Fonksyon_(mat%C3%A9matik)" title="Fonksyon (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Fonksyon (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n" title="Función – 갈리시아어" lang="gl" hreflang="gl" data-title="Función" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה – 히브리어" lang="he" hreflang="he" data-title="פונקציה" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="फलन – 힌디어" lang="hi" hreflang="hi" data-title="फलन" data-language-autonym="हिन्दी" data-language-local-name="힌디어" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Function" title="Function – 피지 힌디어" lang="hif" hreflang="hif" data-title="Function" data-language-autonym="Fiji Hindi" data-language-local-name="피지 힌디어" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – 크로아티아어" lang="hr" hreflang="hr" data-title="Funkcija (matematika)" data-language-autonym="Hrvatski" data-language-local-name="크로아티아어" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/F%C3%BCggv%C3%A9ny_(matematika)" title="Függvény (matematika) – 헝가리어" lang="hu" hreflang="hu" data-title="Függvény (matematika)" data-language-autonym="Magyar" data-language-local-name="헝가리어" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Ֆունկցիա (մաթեմատիկա) – 아르메니아어" lang="hy" hreflang="hy" data-title="Ֆունկցիա (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Function_(mathematica)" title="Function (mathematica) – 인터링구아" lang="ia" hreflang="ia" data-title="Function (mathematica)" data-language-autonym="Interlingua" data-language-local-name="인터링구아" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_(matematika)" title="Fungsi (matematika) – 인도네시아어" lang="id" hreflang="id" data-title="Fungsi (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Funciono" title="Funciono – 이도어" lang="io" hreflang="io" data-title="Funciono" data-language-autonym="Ido" data-language-local-name="이도어" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fall_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Fall (stærðfræði) – 아이슬란드어" lang="is" hreflang="is" data-title="Fall (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="아이슬란드어" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_(matematica)" title="Funzione (matematica) – 이탈리아어" lang="it" hreflang="it" data-title="Funzione (matematica)" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%96%A2%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="関数 (数学) – 일본어" lang="ja" hreflang="ja" data-title="関数 (数学)" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Fongshan_(matimatix)" title="Fongshan (matimatix) – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Fongshan (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/fancu" title="fancu – 로반어" lang="jbo" hreflang="jbo" data-title="fancu" data-language-autonym="La .lojban." data-language-local-name="로반어" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%90_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ფუნქცია (მათემატიკა) – 조지아어" lang="ka" hreflang="ka" data-title="ფუნქცია (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="조지아어" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tas%C9%A3ent_(tusnakt)" title="Tasɣent (tusnakt) – 커바일어" lang="kab" hreflang="kab" data-title="Tasɣent (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="커바일어" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/K%C9%A9lab%C9%A9m" title="Kɩlabɩm – Kabiye" lang="kbp" hreflang="kbp" data-title="Kɩlabɩm" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – 카자흐어" lang="kk" hreflang="kk" data-title="Функция (математика)" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functio" title="Functio – 라틴어" lang="la" hreflang="la" data-title="Functio" data-language-autonym="Latina" data-language-local-name="라틴어" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Funktioun_(Mathematik)" title="Funktioun (Mathematik) – 룩셈부르크어" lang="lb" hreflang="lb" data-title="Funktioun (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="룩셈부르크어" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Fonzion_(matematega)" title="Fonzion (matematega) – Lombard" lang="lmo" hreflang="lmo" data-title="Fonzion (matematega)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%95%E0%BA%B3%E0%BA%A5%E0%BA%B2_(%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94)" title="ຕຳລາ (ຄະນິດສາດ) – 라오어" lang="lo" hreflang="lo" data-title="ຕຳລາ (ຄະນິດສາດ)" data-language-autonym="ລາວ" data-language-local-name="라오어" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – 리투아니아어" lang="lt" hreflang="lt" data-title="Funkcija (matematika)" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Funkcija" title="Funkcija – 라트비아어" lang="lv" hreflang="lv" data-title="Funkcija" data-language-autonym="Latviešu" data-language-local-name="라트비아어" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функција (математика) – 마케도니아어" lang="mk" hreflang="mk" data-title="Функција (математика)" data-language-autonym="Македонски" data-language-local-name="마케도니아어" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AB%E0%B4%99%E0%B5%8D%E0%B4%B7%E0%B5%BB" title="ഫങ്ഷൻ – 말라얄람어" lang="ml" hreflang="ml" data-title="ഫങ്ഷൻ" data-language-autonym="മലയാളം" data-language-local-name="말라얄람어" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA)" title="Функц (математик) – 몽골어" lang="mn" hreflang="mn" data-title="Функц (математик)" data-language-autonym="Монгол" data-language-local-name="몽골어" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AB%E0%A4%B2_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="फल (गणित) – 마라티어" lang="mr" hreflang="mr" data-title="फल (गणित)" data-language-autonym="मराठी" data-language-local-name="마라티어" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fungsi" title="Fungsi – 말레이어" lang="ms" hreflang="ms" data-title="Fungsi" data-language-autonym="Bahasa Melayu" data-language-local-name="말레이어" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Funzjonijiet_(matematika)" title="Funzjonijiet (matematika) – 몰타어" lang="mt" hreflang="mt" data-title="Funzjonijiet (matematika)" data-language-autonym="Malti" data-language-local-name="몰타어" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%96%E1%80%94%E1%80%BA%E1%80%9B%E1%80%BE%E1%80%84%E1%80%BA" title="ဖန်ရှင် – 버마어" lang="my" hreflang="my" data-title="ဖန်ရှင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="버마어" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Afbillen_(Mathematik)" title="Afbillen (Mathematik) – 저지 독일어" lang="nds" hreflang="nds" data-title="Afbillen (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="저지 독일어" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Functie_(wiskunde)" title="Functie (wiskunde) – 네덜란드어" lang="nl" hreflang="nl" data-title="Functie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_funksjon" title="Matematisk funksjon – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Matematisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Funksjon_(matematikk)" title="Funksjon (matematikk) – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Funksjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Aplicacion_(matematicas)" title="Aplicacion (matematicas) – 오크어" lang="oc" hreflang="oc" data-title="Aplicacion (matematicas)" data-language-autonym="Occitan" data-language-local-name="오크어" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Warroomii_(faankishinii)" title="Warroomii (faankishinii) – 오로모어" lang="om" hreflang="om" data-title="Warroomii (faankishinii)" data-language-autonym="Oromoo" data-language-local-name="오로모어" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AB%E0%A9%B0%E0%A8%95%E0%A8%B8%E0%A8%BC%E0%A8%A8_(%E0%A8%B9%E0%A8%BF%E0%A8%B8%E0%A8%BE%E0%A8%AC)" title="ਫੰਕਸ਼ਨ (ਹਿਸਾਬ) – 펀잡어" lang="pa" hreflang="pa" data-title="ਫੰਕਸ਼ਨ (ਹਿਸਾਬ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="펀잡어" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja" title="Funkcja – 폴란드어" lang="pl" hreflang="pl" data-title="Funkcja" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Fonsion" title="Fonsion – Piedmontese" lang="pms" hreflang="pms" data-title="Fonsion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%81%D9%86%DA%A9%D8%B4%D9%86" title="فنکشن – Western Punjabi" lang="pnb" hreflang="pnb" data-title="فنکشن" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Função (matemática) – 포르투갈어" lang="pt" hreflang="pt" data-title="Função (matemática)" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Kinraysuyu" title="Kinraysuyu – 케추아어" lang="qu" hreflang="qu" data-title="Kinraysuyu" data-language-autonym="Runa Simi" data-language-local-name="케추아어" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie" title="Funcție – 루마니아어" lang="ro" hreflang="ro" data-title="Funcție" data-language-autonym="Română" data-language-local-name="루마니아어" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – 러시아어" lang="ru" hreflang="ru" data-title="Функция (математика)" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F._%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%87%D1%8D%D1%80%D1%87%D0%B8%D1%82%D1%8D,_%D1%81%D1%83%D0%BE%D0%BB%D1%82%D0%B0%D0%BB%D0%B0%D1%80%D1%8B%D0%BD_%D1%82%D2%AF%D0%BC%D1%81%D1%8D%D1%8D%D0%BD%D1%8D" title="Функция. Функция чэрчитэ, суолталарын түмсээнэ – 야쿠트어" lang="sah" hreflang="sah" data-title="Функция. Функция чэрчитэ, суолталарын түмсээнэ" data-language-autonym="Саха тыла" data-language-local-name="야쿠트어" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Funzioni_(matim%C3%A0tica)" title="Funzioni (matimàtica) – 시칠리아어" lang="scn" hreflang="scn" data-title="Funzioni (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="시칠리아어" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics) – 스코틀랜드어" lang="sco" hreflang="sco" data-title="Function (mathematics)" data-language-autonym="Scots" data-language-local-name="스코틀랜드어" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Funkcija" title="Funkcija – 세르비아-크로아티아어" lang="sh" hreflang="sh" data-title="Funkcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="세르비아-크로아티아어" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Function_(mathematics)" title="Function (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Function (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Zobrazenie_(matematika)" title="Zobrazenie (matematika) – 슬로바키아어" lang="sk" hreflang="sk" data-title="Zobrazenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="슬로바키아어" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Funkcija_(matematika)" title="Funkcija (matematika) – 슬로베니아어" lang="sl" hreflang="sl" data-title="Funkcija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-smn mw-list-item"><a href="https://smn.wikipedia.org/wiki/Funktio" title="Funktio – 이나리 사미어" lang="smn" hreflang="smn" data-title="Funktio" data-language-autonym="Anarâškielâ" data-language-local-name="이나리 사미어" class="interlanguage-link-target"><span>Anarâškielâ</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Murimo_(Masvomhu)" title="Murimo (Masvomhu) – 쇼나어" lang="sn" hreflang="sn" data-title="Murimo (Masvomhu)" data-language-autonym="ChiShona" data-language-local-name="쇼나어" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Shaqada_(xisaabta)" title="Shaqada (xisaabta) – 소말리아어" lang="so" hreflang="so" data-title="Shaqada (xisaabta)" data-language-autonym="Soomaaliga" data-language-local-name="소말리아어" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Funksioni" title="Funksioni – 알바니아어" lang="sq" hreflang="sq" data-title="Funksioni" data-language-autonym="Shqip" data-language-local-name="알바니아어" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функција (математика) – 세르비아어" lang="sr" hreflang="sr" data-title="Функција (математика)" data-language-autonym="Српски / srpski" data-language-local-name="세르비아어" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Fungsi_(matematika)" title="Fungsi (matematika) – 순다어" lang="su" hreflang="su" data-title="Fungsi (matematika)" data-language-autonym="Sunda" data-language-local-name="순다어" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Funktion" title="Funktion – 스웨덴어" lang="sv" hreflang="sv" data-title="Funktion" data-language-autonym="Svenska" data-language-local-name="스웨덴어" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Funkcyjo" title="Funkcyjo – Silesian" lang="szl" hreflang="szl" data-title="Funkcyjo" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="சார்பு – 타밀어" lang="ta" hreflang="ta" data-title="சார்பு" data-language-autonym="தமிழ்" data-language-local-name="타밀어" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ฟังก์ชัน (คณิตศาสตร์) – 태국어" lang="th" hreflang="th" data-title="ฟังก์ชัน (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="태국어" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Punsiyon_(matematika)" title="Punsiyon (matematika) – 타갈로그어" lang="tl" hreflang="tl" data-title="Punsiyon (matematika)" data-language-autonym="Tagalog" data-language-local-name="타갈로그어" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fonksiyon" title="Fonksiyon – 터키어" lang="tr" hreflang="tr" data-title="Fonksiyon" data-language-autonym="Türkçe" data-language-local-name="터키어" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – 타타르어" lang="tt" hreflang="tt" data-title="Функция (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="타타르어" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-udm mw-list-item"><a href="https://udm.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функция (математика) – 우드말트어" lang="udm" hreflang="udm" data-title="Функция (математика)" data-language-autonym="Удмурт" data-language-local-name="우드말트어" class="interlanguage-link-target"><span>Удмурт</span></a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D9%81%DB%87%D9%86%D9%83%D8%B3%D9%89%D9%8A%DB%95" title="فۇنكسىيە – 위구르어" lang="ug" hreflang="ug" data-title="فۇنكسىيە" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="위구르어" class="interlanguage-link-target"><span>ئۇيغۇرچە / Uyghurche</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Функція (математика) – 우크라이나어" lang="uk" hreflang="uk" data-title="Функція (математика)" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D9%81%D8%A7%D8%B9%D9%84_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="تفاعل (ریاضیات) – 우르두어" lang="ur" hreflang="ur" data-title="تفاعل (ریاضیات)" data-language-autonym="اردو" data-language-local-name="우르두어" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Funksiya_(matematika)" title="Funksiya (matematika) – 우즈베크어" lang="uz" hreflang="uz" data-title="Funksiya (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="우즈베크어" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Funkcii_(matematik)" title="Funkcii (matematik) – Veps" lang="vep" hreflang="vep" data-title="Funkcii (matematik)" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_s%E1%BB%91" title="Hàm số – 베트남어" lang="vi" hreflang="vi" data-title="Hàm số" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Funsiyon_(matematika)" title="Funsiyon (matematika) – 와라이어" lang="war" hreflang="war" data-title="Funsiyon (matematika)" data-language-autonym="Winaray" data-language-local-name="와라이어" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%87%BD%E6%95%B0" title="函数 – 우어" lang="wuu" hreflang="wuu" data-title="函数" data-language-autonym="吴语" data-language-local-name="우어" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Функция – 칼미크어" lang="xal" hreflang="xal" data-title="Функция" data-language-autonym="Хальмг" data-language-local-name="칼미크어" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%A2" title="פונקציע – 이디시어" lang="yi" hreflang="yi" data-title="פונקציע" data-language-autonym="ייִדיש" data-language-local-name="이디시어" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zgh mw-list-item"><a href="https://zgh.wikipedia.org/wiki/%E2%B5%9C%E2%B4%B0%E2%B5%99%E2%B5%96%E2%B5%8F%E2%B5%9C_(%E2%B5%9C%E2%B5%93%E2%B5%99%E2%B5%8F%E2%B4%B0%E2%B4%BD%E2%B5%9C)" title="ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ) – 표준 모로코 타마지트어" lang="zgh" hreflang="zgh" data-title="ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ)" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="표준 모로코 타마지트어" class="interlanguage-link-target"><span>ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%87%BD%E6%95%B0" title="函数 – 중국어" lang="zh" hreflang="zh" data-title="函数" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%98%A0%E5%B0%84" title="映射 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="映射" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/H%C3%A2m-s%C3%B2%CD%98" title="Hâm-sò͘ – 민난어" lang="nan" hreflang="nan" data-title="Hâm-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="민난어" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%87%BD%E6%95%B8" title="函數 – 광둥어" lang="yue" hreflang="yue" data-title="函數" data-language-autonym="粵語" data-language-local-name="광둥어" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> </p> <div class="dablink hatnote"><span typeof="mw:File"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EB%8F%99%EC%9D%8C%EC%9D%B4%EC%9D%98%EC%96%B4_%EB%AC%B8%EC%84%9C" title="위키백과:동음이의어 문서"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/23px-Disambig_grey.svg.png" decoding="async" width="23" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/35px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/46px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span> 다른 뜻에 대해서는 <a href="/wiki/%ED%95%A8%EC%88%98_(%EB%8F%99%EC%9D%8C%EC%9D%B4%EC%9D%98)" class="mw-disambig" title="함수 (동음이의)">함수 (동음이의)</a> 문서를 참고하십시오.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Function_machine2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Function_machine2.svg/220px-Function_machine2.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Function_machine2.svg/330px-Function_machine2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Function_machine2.svg/440px-Function_machine2.svg.png 2x" data-file-width="191" data-file-height="189" /></a><figcaption>함수는 입력값에 따라 출력값을 만들어 내는 ‘블랙 박스’와 같다.</figcaption></figure> <p><a href="/wiki/%EC%88%98%ED%95%99" title="수학">수학</a>에서 <b>함수</b>(函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">function</span>) 또는 <b>사상</b>(寫像, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">map, mapping</span>)은 어떤 <a href="/wiki/%EC%A7%91%ED%95%A9" title="집합">집합</a>의 각 <a href="/wiki/%EC%9B%90%EC%86%8C_(%EC%88%98%ED%95%99)" title="원소 (수학)">원소</a>를 다른 어떤 집합의 유일한 원소에 대응시키는 <a href="/wiki/%EC%9D%B4%ED%95%AD_%EA%B4%80%EA%B3%84" title="이항 관계">이항 관계</a>이다. 대략적으로, 한 <a href="/wiki/%EB%B3%80%EC%88%98_(%EC%88%98%ED%95%99)" title="변수 (수학)">변수</a>의 값에 따라 다른 한 변수의 값이 정해질 때, 후자는 전자의 함수가 된다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Codomain2.SVG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/220px-Codomain2.SVG.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/330px-Codomain2.SVG.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/440px-Codomain2.SVG.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption>함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 정의역 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, 공역 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, 치역 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b884e2d65b3356219702968b6751485fb8f38570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle f(X)}"></span></figcaption></figure> <p><b>함수</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>는 다음과 같은 <a href="/wiki/%ED%8A%9C%ED%94%8C" title="튜플">튜플</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,Y,\operatorname {graph} f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>graph</mi> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,Y,\operatorname {graph} f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca42f5c315fa51e664a00f1ddaf55d6d1e30214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.118ex; height:2.843ex;" alt="{\displaystyle (X,Y,\operatorname {graph} f)}"></span>이다. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>는 <a href="/wiki/%EC%A7%91%ED%95%A9" title="집합">집합</a>이며, 이를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 <b><a href="/wiki/%EC%A0%95%EC%9D%98%EC%97%AD" title="정의역">정의역</a></b>이라고 한다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>는 <a href="/wiki/%EC%A7%91%ED%95%A9" title="집합">집합</a>이며, 이를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 <b><a href="/wiki/%EA%B3%B5%EC%97%AD" title="공역">공역</a></b>이라고 한다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {graph} f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>graph</mi> <mo>⁡</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {graph} f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b47196ac9878d2e86250a4e50aa1741650fa8109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.487ex; height:2.509ex;" alt="{\displaystyle \operatorname {graph} f}"></span>는 <a href="/wiki/%EA%B3%B1%EC%A7%91%ED%95%A9" title="곱집합">곱집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1613c1ff4b6fbfb6c80a8da83e90ad28f0ab3483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.594ex; height:2.176ex;" alt="{\displaystyle X\times Y}"></span>의 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A7%91%ED%95%A9" class="mw-redirect" title="부분 집합">부분 집합</a>이며, 이를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 <b><a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%EA%B7%B8%EB%9E%98%ED%94%84" title="함수의 그래프">그래프</a></b>라고 한다.</li></ul> <p>이 튜플이 다음 조건을 만족시켜야지만 함수라고 한다. </p> <ul><li>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in \operatorname {graph} f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>graph</mi> <mo>⁡</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in \operatorname {graph} f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49aa545a780a580c6b9a5dc4b0852f88660f037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.656ex; height:2.843ex;" alt="{\displaystyle (x,y)\in \operatorname {graph} f}"></span>인 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}"></span>가 유일하게 존재한다. 이러한 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span>라고 쓴다.</li></ul> <p>다시 말해, 함수는 정의역의 각 원소를 정확히 하나의 공역 원소에 대응시킨다. </p><p>표기 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span></dd></dl> <p>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>가 정의역 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, 공역 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>를 갖는 함수라는 뜻이다. 표기 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a052d55c1afcab0a16118128990d5dcaef1abc14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.412ex; height:2.509ex;" alt="{\displaystyle f\colon x\mapsto y}"></span></dd></dl> <p>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5080a8b0a963407ea74ffa50702563771518d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle f(x)=y}"></span>와 같은 뜻이다. </p><p>함수를 정의역과 공역을 생략하여 다음과 같이 표기하기도 한다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\qquad (x\in X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\qquad (x\in X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfd25ba0b2974f97c1a1738627c500c79010608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.022ex; height:2.843ex;" alt="{\displaystyle f(x)\qquad (x\in X)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="예"><span id=".EC.98.88"></span>예</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=2" title="부분 편집: 예"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>만약 어떤 가족의 각 구성원 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span>가 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>의 생년월일이라면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>의 함수가 된다. 이는 각 가족 구성원이 어느 날엔가 태어났고 동시에 두 날에 태어났을 수 없기 때문이다. 이 경우 정의역은 가족 구성원의 집합, 공역은 모든 날짜의 집합으로 취할 수 있다. </p><p>함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \{1,2,3\}\to \{\mathrm {A} ,\mathrm {B} ,\mathrm {C} ,\mathrm {D} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \{1,2,3\}\to \{\mathrm {A} ,\mathrm {B} ,\mathrm {C} ,\mathrm {D} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74365925ee4b376426be5c1aa379b0dc1459b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.076ex; height:2.843ex;" alt="{\displaystyle f\colon \{1,2,3\}\to \{\mathrm {A} ,\mathrm {B} ,\mathrm {C} ,\mathrm {D} \}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 1\mapsto \mathrm {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mn>1</mn> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon 1\mapsto \mathrm {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac4393400a3562f2d9edea5f23a9663780e2aa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.865ex; height:2.509ex;" alt="{\displaystyle f\colon 1\mapsto \mathrm {D} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 2\mapsto \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mn>2</mn> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon 2\mapsto \mathrm {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973ecb4944ea7593dc8fc25d765424dbb4ef1c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.832ex; height:2.509ex;" alt="{\displaystyle f\colon 2\mapsto \mathrm {A} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon 3\mapsto \mathrm {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mn>3</mn> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon 3\mapsto \mathrm {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b843aa1148919a234ab62fdbac4397237a3b8eaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.735ex; height:2.509ex;" alt="{\displaystyle f\colon 3\mapsto \mathrm {B} }"></span></dd></dl> <p>는 정의역이 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,3\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,3\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959905040c4110bae682eae9db986227c5506dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \{1,2,3\}}"></span>, 공역이 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\mathrm {A} ,\mathrm {B} ,\mathrm {C} ,\mathrm {D} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\mathrm {A} ,\mathrm {B} ,\mathrm {C} ,\mathrm {D} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0cf4114226be1da878ae59a8ee8e2dee4209c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.269ex; height:2.843ex;" alt="{\displaystyle \{\mathrm {A} ,\mathrm {B} ,\mathrm {C} ,\mathrm {D} \}}"></span>이며, 1, 2, 3을 각각 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c894024162d89aa8502758b2151cb579806ea9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle \mathrm {D} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6366939c4ebbd4e8494d0dedc54c4b8dd7135a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {A} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93003d072991ba424a73ed1e081afe55c124b6ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.646ex; height:2.176ex;" alt="{\displaystyle \mathrm {B} }"></span>로 대응시키는 이항 관계를 나타낸다. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>가 <a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</a>의 집합이라고 하자. 그렇다면 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e079a08a81cf6846b6d3774c68f2f2c35c98a46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.64ex; height:3.009ex;" alt="{\displaystyle f\colon x\mapsto x^{2}}"></span></dd></dl> <p>는 각 실수를 <a href="/wiki/%EC%A0%9C%EA%B3%B1" title="제곱">제곱</a>시키는 함수이다. 반면, 각 실수에 그보다 큰 실수를 대응시키는 이항 관계는 함수가 아니다. 이는 각 실수보다 큰 실수는 무한히 많으므로 유일하지 않기 때문이다. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2323d9774b2ec5884264fa0e962b1a607b2b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.332ex; height:2.843ex;" alt="{\displaystyle \{c\}}"></span>가 하나의 원소만을 갖는 집합이라고 하자. 그렇다면 임의의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>를 정의역, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2323d9774b2ec5884264fa0e962b1a607b2b6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.332ex; height:2.843ex;" alt="{\displaystyle \{c\}}"></span>를 공역으로 하는 유일한 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to \{c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to \{c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08f00e9bbaf48a2028e97fe25d493b4bb91e269c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.238ex; height:2.843ex;" alt="{\displaystyle f\colon X\to \{c\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8c19cbf3f2a752c2de682019ee22a2b048fac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.263ex; height:2.509ex;" alt="{\displaystyle f\colon x\mapsto c}"></span></dd></dl> <p>가 존재한다. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span>이 <a href="/wiki/%EA%B3%B5%EC%A7%91%ED%95%A9" title="공집합">공집합</a>이라고 하자. 그렇다면 임의의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00595c5e33692e724937fdcc8870496acce1ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }"></span>을 정의역, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>를 공역으로 하는 유일한 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \varnothing \to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi class="MJX-variant">∅</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \varnothing \to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa7fbe3b549bfbd3bf24ee7b26a50092f3646c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.508ex; height:2.509ex;" alt="{\displaystyle f\colon \varnothing \to Y}"></span></dd></dl> <p>가 존재한다. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628ddbc2e88522dbee42a7c3c3390d1f464f571a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.68ex; height:2.176ex;" alt="{\displaystyle Y=\varnothing }"></span>일 경우 이는 공역이 공집합인 유일한 함수이다. </p> <div class="mw-heading mw-heading2"><h2 id="종류"><span id=".EC.A2.85.EB.A5.98"></span>종류</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=3" title="부분 편집: 종류"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="단사_함수와_전사_함수"><span id=".EB.8B.A8.EC.82.AC_.ED.95.A8.EC.88.98.EC.99.80_.EC.A0.84.EC.82.AC_.ED.95.A8.EC.88.98"></span>단사 함수와 전사 함수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=4" title="부분 편집: 단사 함수와 전사 함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r34311305">.mw-parser-output .hatnote{}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="단사 함수">단사 함수</a>, <a href="/wiki/%EC%A0%84%EC%82%AC_%ED%95%A8%EC%88%98" title="전사 함수">전사 함수</a> 및 <a href="/wiki/%EC%A0%84%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="전단사 함수">전단사 함수</a>입니다.</div> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 122px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Injection.svg" class="mw-file-description" title="단사 함수의 예"><img alt="단사 함수의 예" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Injection.svg/180px-Injection.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Injection.svg/270px-Injection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Injection.svg/360px-Injection.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">단사 함수의 예</div> </li> <li class="gallerybox" style="width: 122px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Surjection.svg" class="mw-file-description" title="전사 함수의 예"><img alt="전사 함수의 예" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Surjection.svg/180px-Surjection.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Surjection.svg/270px-Surjection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Surjection.svg/360px-Surjection.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">전사 함수의 예</div> </li> <li class="gallerybox" style="width: 122px"> <div class="thumb" style="width: 120px;"><span typeof="mw:File"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Bijection.svg" class="mw-file-description" title="전단사 함수의 예"><img alt="전단사 함수의 예" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/180px-Bijection.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/270px-Bijection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bijection.svg/360px-Bijection.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">전단사 함수의 예</div> </li> </ul> <p>함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>에 대하여, 다음과 같은 성질들을 정의할 수 있다. </p> <ul><li><b><a href="/wiki/%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="단사 함수">단사 함수</a></b>: 임의의 정의역 원소 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}"></span>에 대하여, 만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=f(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=f(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c45b4d507fbe420cf82e042d6f32f7aed7f8a12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f(x)=f(y)}"></span>라면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}"></span>이다. 즉, 서로 다른 정의역 원소는 서로 다른 공역 원소에 대응한다.<sup id="cite_ref-이상구_1-0" class="reference"><a href="#cite_note-이상구-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><b><a href="/wiki/%EC%A0%84%EC%82%AC_%ED%95%A8%EC%88%98" title="전사 함수">전사 함수</a></b>: 임의의 공역 원소 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span>인 정의역 원소 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>가 존재한다. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 <a href="/wiki/%EC%B9%98%EC%97%AD" title="치역">치역</a>은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 공역과 같다.<sup id="cite_ref-이상구_1-1" class="reference"><a href="#cite_note-이상구-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li><b><a href="/wiki/%EC%A0%84%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="전단사 함수">전단사 함수</a></b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>는 단사 함수이며, 전사 함수이다. 이는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>가 <a href="/wiki/%EC%97%AD%ED%95%A8%EC%88%98" title="역함수">역함수</a>를 갖는 것과 동치이다.<sup id="cite_ref-이상구_1-2" class="reference"><a href="#cite_note-이상구-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="특별한_정의역·공역을_갖는_함수"><span id=".ED.8A.B9.EB.B3.84.ED.95.9C_.EC.A0.95.EC.9D.98.EC.97.AD.C2.B7.EA.B3.B5.EC.97.AD.EC.9D.84_.EA.B0.96.EB.8A.94_.ED.95.A8.EC.88.98"></span>특별한 정의역·공역을 갖는 함수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=5" title="부분 편집: 특별한 정의역·공역을 갖는 함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Complex_gamma.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Complex_gamma.jpg/220px-Complex_gamma.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Complex_gamma.jpg/330px-Complex_gamma.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Complex_gamma.jpg/440px-Complex_gamma.jpg 2x" data-file-width="731" data-file-height="731" /></a><figcaption><a href="/wiki/%EA%B0%90%EB%A7%88_%ED%95%A8%EC%88%98" title="감마 함수">감마 함수</a>의 그래프</figcaption></figure> <p>특별한 정의역 또는 공역을 갖는 함수는 특별한 이름이 붙는다. </p> <table class="wikitable"> <tbody><tr> <th>정의역</th> <th>공역</th> <th>이름 </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></td> <td></td> <td><b><a href="/wiki/%EC%88%98%EC%97%B4" title="수열">수열</a></b> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>의 <a href="/wiki/%EC%97%B4%EB%A6%B0%EC%A7%91%ED%95%A9" title="열린집합">열린집합</a></td> <td></td> <td><b>실변수 함수</b>(實變數函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">function of a real variable</span>) </td></tr> <tr> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span></td> <td><b>실숫값 함수</b>(實數-函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">real-valued function</span>) </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>의 열린집합</td> <td></td> <td><b>복소변수 함수</b>(複素變數函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">function of a complex variable</span>) </td></tr> <tr> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span></td> <td><b>복소값 함수</b>(複素-函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">complex-valued function</span>) </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> (또는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span>)의 열린집합</td> <td></td> <td><b>다변수 함수</b>(多變數函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">multivariate function</span>) </td></tr></tbody></table> <p>특별한 정의역을 갖는 함수에 대하여 추가적인 성질들을 정의할 수 있다. 예컨대 두 <a href="/wiki/%EC%9C%84%EC%83%81_%EA%B3%B5%EA%B0%84_(%EC%88%98%ED%95%99)" title="위상 공간 (수학)">위상 공간</a> 사이의 함수에 대하여 <a href="/wiki/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수">연속 함수</a>의 개념을 정의할 수 있으며, 두 <a href="/wiki/%EB%A7%A4%EB%81%84%EB%9F%AC%EC%9A%B4_%EB%8B%A4%EC%96%91%EC%B2%B4" title="매끄러운 다양체">매끄러운 다양체</a> 사이의 함수의 각종 매끄러움 성질들을 정의할 수 있다. 실변수 실숫값 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon U\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon U\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d71097bf5f00c4116453f42be6ed7f74e74b3be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.387ex; height:2.509ex;" alt="{\displaystyle f\colon U\to \mathbb {R} }"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>의 <a href="/wiki/%EC%97%B4%EB%A6%B0%EC%A7%91%ED%95%A9" title="열린집합">열린집합</a>)의 경우 추가로 다음과 같은 성질들을 정의할 수 있다. </p> <ul><li><a href="/wiki/%EB%8B%A8%EC%A1%B0%ED%95%A8%EC%88%98" title="단조함수">단조함수</a><sup id="cite_ref-이상구_1-3" class="reference"><a href="#cite_note-이상구-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <ul><li><a href="/wiki/%EC%A6%9D%EA%B0%80%ED%95%A8%EC%88%98" class="mw-redirect" title="증가함수">증가함수</a>: 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/714d2ffe0940e25a39303f99980be51dd55b2257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.142ex; height:2.509ex;" alt="{\displaystyle x,y\in U}"></span>에 대하여, 만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span>라면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\leq f(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\leq f(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef64a0cfa8c32bf9d3f0549cb5ab1b3f46af5dd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f(x)\leq f(y)}"></span>. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 <a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%EA%B7%B8%EB%9E%98%ED%94%84" title="함수의 그래프">그래프</a>는 오른쪽으로 갈수록 상승하는 곡선이다. 예를 들어, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7144f67be80ffe9b4b8f4d09cde56282d86443f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U=\mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto 2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto 2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e3f84003b9d1581877cb8c113b67dee71cb641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.436ex; height:2.176ex;" alt="{\displaystyle x\mapsto 2x}"></span></li> <li><a href="/wiki/%EA%B0%90%EC%86%8C%ED%95%A8%EC%88%98" class="mw-redirect" title="감소함수">감소함수</a>: 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/714d2ffe0940e25a39303f99980be51dd55b2257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.142ex; height:2.509ex;" alt="{\displaystyle x,y\in U}"></span>에 대하여, 만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span>라면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\geq f(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\geq f(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3f82acd1f762cb40b78323f68c0b3c64cb4cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f(x)\geq f(y)}"></span>. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 그래프는 오른쪽으로 갈수록 하강한다. 예를 들어, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7144f67be80ffe9b4b8f4d09cde56282d86443f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U=\mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto -2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto -2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76ec921c6edd251a78573a77231ccc91c23acbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.244ex; height:2.343ex;" alt="{\displaystyle x\mapsto -2x}"></span></li></ul></li> <li><a href="/wiki/%ED%99%80%ED%95%A8%EC%88%98%EC%99%80_%EC%A7%9D%ED%95%A8%EC%88%98" title="홀함수와 짝함수">홀함수와 짝함수</a><sup id="cite_ref-박은순_2-0" class="reference"><a href="#cite_note-박은순-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:130</sup></span> <ul><li><a href="/wiki/%ED%99%80%ED%95%A8%EC%88%98" class="mw-redirect" title="홀함수">홀함수</a>: 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0a31cf0a2e69a1dc94e0d2a910591ce6574d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.761ex; height:2.343ex;" alt="{\displaystyle -x\in U}"></span>이며 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)=-f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-x)=-f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b022ffe516cf5bc26a68fd954753aa2bddf508f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.55ex; height:2.843ex;" alt="{\displaystyle f(-x)=-f(x)}"></span>. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 그래프는 원점에 대하여 대칭이다. 예를 들어, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7144f67be80ffe9b4b8f4d09cde56282d86443f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U=\mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e683c4c1189ad0c5332cf442b33f1cfe8fc3996d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.328ex; height:2.676ex;" alt="{\displaystyle x\mapsto x^{3}}"></span></li> <li><a href="/wiki/%EC%A7%9D%ED%95%A8%EC%88%98" class="mw-redirect" title="짝함수">짝함수</a>: 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0a31cf0a2e69a1dc94e0d2a910591ce6574d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.761ex; height:2.343ex;" alt="{\displaystyle -x\in U}"></span>이며 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-x)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185fd2e78903788bc5756b067d0ac6aae1846724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.742ex; height:2.843ex;" alt="{\displaystyle f(-x)=f(x)}"></span>. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 그래프는 y축에 대하여 대칭이다. 예를 들어, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7144f67be80ffe9b4b8f4d09cde56282d86443f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U=\mathbb {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b49de3850ec5b2be3acb8db45514958c5e80ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.328ex; height:2.676ex;" alt="{\displaystyle x\mapsto x^{2}}"></span></li></ul></li> <li><a href="/wiki/%EC%A3%BC%EA%B8%B0_%ED%95%A8%EC%88%98" class="mw-redirect" title="주기 함수">주기 함수</a>: 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+T\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>T</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+T\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb3e55d6da68c19f15df3e22932c904af877fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.43ex; height:2.343ex;" alt="{\displaystyle x+T\in U}"></span>이며 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+T)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+T)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d267d80375326ebbc3c4efdc9d22a2feaa9bb325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.41ex; height:2.843ex;" alt="{\displaystyle f(x+T)=f(x)}"></span>. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 그래프는 x축 방향의 평행 이동 대칭을 갖는다. 대표적으로 모든 <a href="/wiki/%EC%82%BC%EA%B0%81_%ED%95%A8%EC%88%98" title="삼각 함수">삼각 함수</a>는 주기 함수다.</li></ul> <div class="mw-heading mw-heading3"><h3 id="조각마다_정의된_함수"><span id=".EC.A1.B0.EA.B0.81.EB.A7.88.EB.8B.A4_.EC.A0.95.EC.9D.98.EB.90.9C_.ED.95.A8.EC.88.98"></span>조각마다 정의된 함수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=6" title="부분 편집: 조각마다 정의된 함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Upper_semi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Upper_semi.svg/220px-Upper_semi.svg.png" decoding="async" width="220" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Upper_semi.svg/330px-Upper_semi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Upper_semi.svg/440px-Upper_semi.svg.png 2x" data-file-width="214" data-file-height="153" /></a><figcaption>이 함수는 불연속점 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>을 가지지만 각 구간 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-\infty ,x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-\infty ,x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23b9fc728b90c2de60b2d11b6b224ff0da55ec5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.101ex; height:2.843ex;" alt="{\displaystyle [-\infty ,x_{0})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{0},\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{0},\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09bb2601e27d029931e764b386b3ad15938519fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.293ex; height:2.843ex;" alt="{\displaystyle [x_{0},\infty )}"></span>에서 매끄럽다.</figcaption></figure> <p>두 <a href="/wiki/%EB%A7%A4%EB%81%84%EB%9F%AC%EC%9A%B4_%EB%8B%A4%EC%96%91%EC%B2%B4" title="매끄러운 다양체">매끄러운 다양체</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}"></span> 사이의 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>에 대하여, 다음 두 조건을 만족시키는 유한 개의 서로소 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dotsc ,X_{n}\subset X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊂</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dotsc ,X_{n}\subset X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6e3c4dc47799fc7edb942e1482d457cc65665e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.378ex; height:2.509ex;" alt="{\displaystyle X_{1},\dotsc ,X_{n}\subset X}"></span>이 존재한다면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>를 <b>조각마다 〜 함수</b>라고 한다. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=X_{1}\cup \cdots \cup X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∪</mo> <mo>⋯</mo> <mo>∪</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=X_{1}\cup \cdots \cup X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250b763d61172030c2a4cfec669a6dc6cad940d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.088ex; height:2.509ex;" alt="{\displaystyle X=X_{1}\cup \cdots \cup X_{n}}"></span></li> <li>각 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dotsc ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dotsc ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f2132430a61b900cf2c4380774394ca9f09c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\dotsc ,n}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}\subseteq X_{i}\subseteq \operatorname {cl} D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊆</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊆</mo> <mi>cl</mi> <mo>⁡</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}\subseteq X_{i}\subseteq \operatorname {cl} D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24db59f1045302a2203afdbc78bca3b91d6a29b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.435ex; height:2.509ex;" alt="{\displaystyle D_{i}\subseteq X_{i}\subseteq \operatorname {cl} D_{i}}"></span>인 <a href="/wiki/%EC%98%81%EC%97%AD_(%ED%95%B4%EC%84%9D%ED%95%99)" title="영역 (해석학)">영역</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6332ad38f9beb28b133a558016e19415de0b8cef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.802ex; height:2.509ex;" alt="{\displaystyle D_{i}\subseteq X}"></span>가 존재한다.</li> <li>각 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dotsc ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dotsc ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f2132430a61b900cf2c4380774394ca9f09c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\displaystyle i=1,\dotsc ,n}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\upharpoonright X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↾</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\upharpoonright X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed82ce3eeabaecde16f581d0f3b2fb9bfb9680e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.263ex; height:2.509ex;" alt="{\displaystyle f\upharpoonright X_{i}}"></span>는 〜 함수이다.</li></ul> <p>특히, 정의역이 실수 <a href="/wiki/%EA%B5%AC%EA%B0%84" title="구간">구간</a>인 경우, 정의역은 작은 구간들로 분할되어야 한다. 예를 들어, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042e529c93e86b5d03b1e1cd6ddcc50e89761c03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.429ex; height:2.843ex;" alt="{\displaystyle [0,1]\subseteq \mathbb {R} }"></span>의 분할의 한 가지 예는 다음과 같다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]=[0,1/4)\sqcup [1/4,1/2)\sqcup \{1/2\}\sqcup [1/2,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">)</mo> <mo>⊔</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⊔</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>⊔</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]=[0,1/4)\sqcup [1/4,1/2)\sqcup \{1/2\}\sqcup [1/2,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c134d9f3f95d7195e98374ec4c0db385c72e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.084ex; height:2.843ex;" alt="{\displaystyle [0,1]=[0,1/4)\sqcup [1/4,1/2)\sqcup \{1/2\}\sqcup [1/2,1]}"></span></dd></dl> <p>예를 들어, 실수 <a href="/wiki/%EC%A0%88%EB%8C%93%EA%B0%92" title="절댓값">절댓값</a> 함수는 조각마다 <a href="/wiki/%EC%9D%BC%EC%B0%A8_%ED%95%A8%EC%88%98" title="일차 함수">일차 함수</a>이다. <a href="/wiki/%EB%B6%80%ED%98%B8_%ED%95%A8%EC%88%98" class="mw-redirect" title="부호 함수">부호 함수</a>는 조각마다 <a href="/wiki/%EC%83%81%EC%88%98_%ED%95%A8%EC%88%98" title="상수 함수">상수 함수</a>이다. 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto {\begin{cases}x^{2}&x<1/2\\-x^{2}+2x&x\geq 1/2\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi>x</mi> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mi>x</mi> <mo>≥</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto {\begin{cases}x^{2}&x<1/2\\-x^{2}+2x&x\geq 1/2\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58dcbcd67e6a5d209055631dc7c8cf26fa8ad40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.514ex; height:6.176ex;" alt="{\displaystyle f\colon x\mapsto {\begin{cases}x^{2}&x<1/2\\-x^{2}+2x&x\geq 1/2\end{cases}}}"></span></dd></dl> <p>는 조각마다 <a href="/wiki/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수">연속 함수</a>이다. </p> <div class="mw-heading mw-heading3"><h3 id="다가_함수"><span id=".EB.8B.A4.EA.B0.80_.ED.95.A8.EC.88.98"></span>다가 함수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=7" title="부분 편집: 다가 함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%8B%A4%EA%B0%80_%ED%95%A8%EC%88%98" title="다가 함수">다가 함수</a>입니다.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> <a href="/wiki/%EB%B6%84%EC%A7%80_%EC%A0%88%EB%8B%A8" class="mw-redirect" title="분지 절단">분지 절단</a> 문서를 참고하십시오.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> <a href="/wiki/%EC%9D%8C%ED%95%A8%EC%88%98_%EC%A0%95%EB%A6%AC" title="음함수 정리">음함수 정리</a> 문서를 참고하십시오.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Function_with_two_values_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Function_with_two_values_1.svg/220px-Function_with_two_values_1.svg.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Function_with_two_values_1.svg/330px-Function_with_two_values_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Function_with_two_values_1.svg/440px-Function_with_two_values_1.svg.png 2x" data-file-width="440" data-file-height="320" /></a><figcaption>다가 함수의 예</figcaption></figure> <p><a href="/wiki/%EB%8B%A4%EA%B0%80_%ED%95%A8%EC%88%98" title="다가 함수">다가 함수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>의 정의는 함수의 조건을 다음과 같이 약화시켜 얻는다. </p> <ul><li>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\in \operatorname {graph} f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>graph</mi> <mo>⁡</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in \operatorname {graph} f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49aa545a780a580c6b9a5dc4b0852f88660f037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.656ex; height:2.843ex;" alt="{\displaystyle (x,y)\in \operatorname {graph} f}"></span>인 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}"></span>가 적어도 하나 존재한다.</li></ul> <p>즉, 다가 함수는 정의역의 각 원소를 적어도 하나의 공역 원소에 대응시키지만, 함수와 달리 여러 개의 공역 원소에 대응시킬 수 있다. 다가 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>는 일반적으로 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>에서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>로 가는 함수가 아니지만 <a href="/wiki/%EB%A9%B1%EC%A7%91%ED%95%A9" title="멱집합">멱집합</a>으로 가는 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to {\mathcal {P}}(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to {\mathcal {P}}(Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/368b16f7a2ef85b01193fe313fe92c3abe6c03ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.193ex; height:2.843ex;" alt="{\displaystyle f\colon X\to {\mathcal {P}}(Y)}"></span></dd></dl> <p>와 동치이다. </p><p><a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</a>의 <a href="/wiki/%EA%B1%B0%EB%93%AD%EC%A0%9C%EA%B3%B1" title="거듭제곱">거듭제곱</a>은 대표적인 다가 함수이다. 특히 음이 아닌 실수의 <a href="/wiki/%EC%A0%9C%EA%B3%B1%EA%B7%BC" title="제곱근">제곱근</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon [0,\infty )\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon [0,\infty )\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912ac1476c4c608cb533e2833e4e98b2b712936b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.676ex; height:2.843ex;" alt="{\displaystyle f\colon [0,\infty )\to \mathbb {R} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto \pm {\sqrt {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto \pm {\sqrt {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1433da3f91bc4bd4242ea6dade85c09174ffdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.33ex; height:3.009ex;" alt="{\displaystyle f\colon x\mapsto \pm {\sqrt {x}}}"></span></dd></dl> <p>은 (양의 실수가 두 개의 제곱근을 가지므로) 다가 함수이다. </p><p>일반적인 함수를 다가 함수와 구별하기 위해 <b>일가 함수</b>(一價函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">single-valued function</span>)라고 부르기도 한다.<sup id="cite_ref-이상구_1-4" class="reference"><a href="#cite_note-이상구-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="부분_정의_함수"><span id=".EB.B6.80.EB.B6.84_.EC.A0.95.EC.9D.98_.ED.95.A8.EC.88.98"></span>부분 정의 함수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=8" title="부분 편집: 부분 정의 함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A0%95%EC%9D%98_%ED%95%A8%EC%88%98" title="부분 정의 함수">부분 정의 함수</a>입니다.</div> <p><a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A0%95%EC%9D%98_%ED%95%A8%EC%88%98" title="부분 정의 함수">부분 정의 함수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>는 다음과 같이 약화된 조건을 사용하여 정의한다. </p> <ul><li>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span> 및 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y,z\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y,z\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b275c540da410eb72fd18765f95545f28d10abbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.892ex; height:2.509ex;" alt="{\displaystyle y,z\in Y}"></span>에 대하여, 만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y),(x,z)\in \operatorname {graph} f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>graph</mi> <mo>⁡</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y),(x,z)\in \operatorname {graph} f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0475a42b9f5bbe1d707c7595bc121224dae777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.951ex; height:2.843ex;" alt="{\displaystyle (x,y),(x,z)\in \operatorname {graph} f}"></span>라면 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de8f5225200d63f8f4678bbe6d07c4d384e73ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.009ex;" alt="{\displaystyle y=z}"></span>이다.</li></ul> <p>즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>의 각 원소는 유일한 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>의 원소에 대응하거나, 어떤 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>의 원소에도 대응하지 않는다. 부분 정의 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>는 일반적으로 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>에서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>로 가는 함수가 아니다. 그러나 이는 다음과 같은 꼴의 함수와 동치이다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\cup \{\bullet _{X}\}\to Y\cup \{\bullet _{Y}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mo>∙</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">→</mo> <mi>Y</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mo>∙</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\cup \{\bullet _{X}\}\to Y\cup \{\bullet _{Y}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/541873c5ce29979bea5d1b5de1bea2d3f1156899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.938ex; height:2.843ex;" alt="{\displaystyle f\colon X\cup \{\bullet _{X}\}\to Y\cup \{\bullet _{Y}\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\bullet _{X})=\bullet _{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mo>∙</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∙</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\bullet _{X})=\bullet _{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe84e916fb4b2181362c50ad4e6e4afe54a5522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.63ex; height:2.843ex;" alt="{\displaystyle f(\bullet _{X})=\bullet _{Y}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bullet _{X}\not \in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∙</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>∉</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bullet _{X}\not \in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c50eb974e0a12a9bb43a8b8ebc770ddec8f41d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.615ex; height:2.676ex;" alt="{\displaystyle \bullet _{X}\not \in X}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bullet _{Y}\not \in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∙</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>∉</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bullet _{Y}\not \in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1593f3e1e258b3d9ed615688a9d397e1370eae32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.263ex; height:2.676ex;" alt="{\displaystyle \bullet _{Y}\not \in Y}"></span></dd></dl> <p>예를 들어, 역수를 취하는 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto {\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9cf37fef945717888ce2442bee8485933c97487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.422ex; height:5.176ex;" alt="{\displaystyle f\colon x\mapsto {\frac {1}{x}}}"></span></dd></dl> <p>는 (0의 역수가 정의되지 않으므로) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> 위의 부분 정의 함수이다. 이 부분 정의 함수는 정의역을 0이 아닌 실수로 축소하거나 공역에 (음과 양을 구분하지 않는) 무한대 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\infty }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∞</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\infty }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ea7c1eeb0ff8cabf984488a8f8eb299dc51cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.028ex; margin-left: -0.01ex; margin-right: -0.009ex; margin-bottom: -0.31ex; width:2.343ex; height:2.343ex;" alt="{\displaystyle {\widehat {\infty }}}"></span>를 추가하여 함수로 만들 수 있다. </p><p>일반적인 함수를 부분 정의 함수와 구별하기 위해 <b>전함수</b>(全函數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">total function</span>)라고 부르기도 한다. </p> <div class="mw-heading mw-heading2"><h2 id="연산"><span id=".EC.97.B0.EC.82.B0"></span>연산</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=9" title="부분 편집: 연산"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="상과_원상"><span id=".EC.83.81.EA.B3.BC_.EC.9B.90.EC.83.81"></span>상과 원상</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=10" title="부분 편집: 상과 원상"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%83%81_(%EC%88%98%ED%95%99)" title="상 (수학)">상 (수학)</a>입니다.</div> <p>집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dce86da0107830a9a97287f9486d9b4ff022875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.822ex; height:2.343ex;" alt="{\displaystyle A\subseteq X}"></span> 및 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>에 대하여, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f(a)\colon a\in A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>a</mi> <mo>∈</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f(a)\colon a\in A\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f23649e21594a678ba71508bbe9a41a39bcbc396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.49ex; height:2.843ex;" alt="{\displaystyle \{f(a)\colon a\in A\}}"></span></dd></dl> <p>를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>의 <b>상</b>이라고 하며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2711c647e5397c7016ee21bbcea53565480bd5e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.831ex; height:2.843ex;" alt="{\displaystyle f(A)}"></span>로 쓴다. 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⊆</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\subseteq Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ee25ef2534c4c67e0c9b5a05a4f2770d1e1db6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.636ex; height:2.343ex;" alt="{\displaystyle B\subseteq Y}"></span> 및 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>에 대하여, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in X\colon f(x)\in B\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\in X\colon f(x)\in B\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85f78b2c691b481d84278ab91e77a1545683ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.531ex; height:2.843ex;" alt="{\displaystyle \{x\in X\colon f(x)\in B\}}"></span></dd></dl> <p>를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>의 <b>원상</b>이라고 하며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/572ddad8cd0a0758fb98e1c94c432dc2f7a06636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.227ex; height:3.176ex;" alt="{\displaystyle f^{-1}(B)}"></span>로 쓴다. 정의역의 상 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b884e2d65b3356219702968b6751485fb8f38570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle f(X)}"></span>을 <b><a href="/wiki/%EC%B9%98%EC%97%AD" title="치역">치역</a></b>이라고 한다. </p><p>예를 들어, <a href="/wiki/%EC%82%AC%EC%9D%B8_%ED%95%A8%EC%88%98" class="mw-redirect" title="사인 함수">사인 함수</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \colon \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \colon \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a777514d4123261aaf129396161f2bbd2a3f53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.86ex; height:2.176ex;" alt="{\displaystyle \sin \colon \mathbb {R} \to \mathbb {R} }"></span></dd></dl> <p>의 치역은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"></span>이며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{1\}}"></span>의 원상은 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\left(2n+{\frac {1}{2}}\right)\pi \colon n\in \mathbb {Z} \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>π</mi> <mo>:</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\left(2n+{\frac {1}{2}}\right)\pi \colon n\in \mathbb {Z} \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d2d6d0db274b89d36b18d6f2d2c7d766490955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.842ex; height:6.176ex;" alt="{\displaystyle \left\{\left(2n+{\frac {1}{2}}\right)\pi \colon n\in \mathbb {Z} \right\}}"></span></dd></dl> <p>이다. 여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>는 <a href="/wiki/%EC%A0%95%EC%88%98" title="정수">정수</a>의 집합, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>는 <a href="/wiki/%EC%9B%90%EC%A3%BC%EC%9C%A8" title="원주율">원주율</a>이다. </p> <div class="mw-heading mw-heading3"><h3 id="역함수"><span id=".EC.97.AD.ED.95.A8.EC.88.98"></span>역함수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=11" title="부분 편집: 역함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%97%AD%ED%95%A8%EC%88%98" title="역함수">역함수</a>입니다.</div> <p>함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>에 대하여, 다음 두 조건이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. </p> <ul><li>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1c0ec36a82f33f5e3d7434d5667881b4ec323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span>인 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>가 유일하게 존재한다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>는 <a href="/wiki/%EC%A0%84%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="전단사 함수">전단사 함수</a>이다.</li></ul> <p>이 경우 <a href="/wiki/%EC%A0%95%EC%9D%98%EC%97%AD" title="정의역">정의역</a>과 <a href="/wiki/%EA%B3%B5%EC%97%AD" title="공역">공역</a>이 뒤바뀌고 대응의 방향이 반대로 바뀐 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\colon Y\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}\colon Y\to X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a1bc0edf199414feff53da55c19b265bc5015a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.055ex; height:3.009ex;" alt="{\displaystyle f^{-1}\colon Y\to X}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\colon f(x)\mapsto x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}\colon f(x)\mapsto x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f3ac9c7656e45efa031885af023780c905e296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.049ex; height:3.176ex;" alt="{\displaystyle f^{-1}\colon f(x)\mapsto x}"></span></dd></dl> <p>를 생각할 수 있다. 이를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 <b><a href="/wiki/%EC%97%AD%ED%95%A8%EC%88%98" title="역함수">역함수</a></b>라고 한다. </p><p>예를 들어, <a href="/wiki/%EC%A7%80%EC%88%98_%ED%95%A8%EC%88%98" title="지수 함수">지수 함수</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \colon \mathbb {R} \to (0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \colon \mathbb {R} \to (0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30599adb8a284c1ea8c4518bf57a9fe57d1a2ca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.208ex; height:2.843ex;" alt="{\displaystyle \exp \colon \mathbb {R} \to (0,\infty )}"></span></dd></dl> <p>는 전단사 함수이며, 그 역함수는 <a href="/wiki/%EB%A1%9C%EA%B7%B8_%ED%95%A8%EC%88%98" class="mw-redirect" title="로그 함수">로그 함수</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \colon (0,\infty )\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \colon (0,\infty )\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/097f4f3e5cec211d85a0aad740ba6d95fc1faba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.595ex; height:2.843ex;" alt="{\displaystyle \ln \colon (0,\infty )\to \mathbb {R} }"></span></dd></dl> <p>이다. </p> <div class="mw-heading mw-heading3"><h3 id="합성"><span id=".ED.95.A9.EC.84.B1"></span>합성</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=12" title="부분 편집: 합성"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%ED%95%A9%EC%84%B1" title="함수의 합성">함수의 합성</a>입니다.</div> <p>첫째 함수의 공역과 둘째 함수의 정의역이 같은 두 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span> 및 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon Y\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon Y\to Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6268a0bd7f3016093edf824725f1ffcba89f8064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.218ex; height:2.509ex;" alt="{\displaystyle g\colon Y\to Z}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>의 원소를 먼저 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>를 통해 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>의 원소에 대응시키고, 다시 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>에 따라 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>의 원소로 대응시키는 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f\colon X\to Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f\colon X\to Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842eb8ad3cc36e0f1b71d62b7bb90bdbe6c79091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.898ex; height:2.509ex;" alt="{\displaystyle g\circ f\colon X\to Z}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f\colon x\mapsto g(f(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f\colon x\mapsto g(f(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11336ff70938a86a1e12320630524c59012178a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.91ex; height:2.843ex;" alt="{\displaystyle g\circ f\colon x\mapsto g(f(x))}"></span></dd></dl> <p>를 생각할 수 있다. 이를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>와 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>의 <b><a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%ED%95%A9%EC%84%B1" title="함수의 합성">합성</a></b>이라고 한다. </p><p>예를 들어, 만약 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=Y=Z=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>Y</mi> <mo>=</mo> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=Y=Z=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0641fb232ec360f0875bd1a7a052b249968ef688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.407ex; height:2.176ex;" alt="{\displaystyle X=Y=Z=\mathbb {R} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto 2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto 2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26760fa15f21d7beefcc6819ca55cd74f7db1c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.748ex; height:2.509ex;" alt="{\displaystyle f\colon x\mapsto 2x}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon y\mapsto y+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>y</mi> <mo stretchy="false">↦</mo> <mi>y</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\colon y\mapsto y+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c559a74f0e64993dab4cc580f01a37bc1a97bce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.078ex; height:2.509ex;" alt="{\displaystyle g\colon y\mapsto y+1}"></span></dd></dl> <p>일 경우 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f\colon x\mapsto 2x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\circ f\colon x\mapsto 2x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918a0c6d5f1d74ae39340a1e7f6021a01091984b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.062ex; height:2.509ex;" alt="{\displaystyle g\circ f\colon x\mapsto 2x+1}"></span></dd></dl> <p>이다. </p> <div class="mw-heading mw-heading3"><h3 id="제한"><span id=".EC.A0.9C.ED.95.9C"></span>제한</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=13" title="부분 편집: 제한"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%A0%9C%ED%95%9C_(%EC%88%98%ED%95%99)" title="제한 (수학)">제한 (수학)</a>입니다.</div> <p>함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></span>의 정의역의 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A7%91%ED%95%A9" class="mw-redirect" title="부분 집합">부분 집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dce86da0107830a9a97287f9486d9b4ff022875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.822ex; height:2.343ex;" alt="{\displaystyle A\subseteq X}"></span>으로의 <b>제한</b>(制限, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">restriction</span>)은 다음과 같은 함수를 일컫는다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\upharpoonright A\colon A\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↾</mo> <mi>A</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\upharpoonright A\colon A\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56ba5fd2fdfe9889ce16a78e4215e89b7b53f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.446ex; height:2.509ex;" alt="{\displaystyle f\upharpoonright A\colon A\to Y}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\upharpoonright A\colon x\mapsto f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↾</mo> <mi>A</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\upharpoonright A\colon x\mapsto f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535c6a0299377bc6fa2f3458e791794a9e639c7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.677ex; height:2.843ex;" alt="{\displaystyle f\upharpoonright A\colon x\mapsto f(x)}"></span></dd></dl> <p>즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 대응 규칙을 유지한 채 정의역만 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>로 줄인 함수이다. </p> <div class="mw-heading mw-heading3"><h3 id="호환되는_함수족으로_유도되는_함수"><span id=".ED.98.B8.ED.99.98.EB.90.98.EB.8A.94_.ED.95.A8.EC.88.98.EC.A1.B1.EC.9C.BC.EB.A1.9C_.EC.9C.A0.EB.8F.84.EB.90.98.EB.8A.94_.ED.95.A8.EC.88.98"></span>호환되는 함수족으로 유도되는 함수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=14" title="부분 편집: 호환되는 함수족으로 유도되는 함수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>함수족 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i}\colon X_{i}\to Y_{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i}\colon X_{i}\to Y_{i})_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/401002250da320e1f5c96a25f8452d23156f55a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.995ex; height:2.843ex;" alt="{\displaystyle (f_{i}\colon X_{i}\to Y_{i})_{i\in I}}"></span>가 다음 조건을 만족시킨다고 하자. </p> <ul><li>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7596675a16b3250fc6b02be961d315d3ed467af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.807ex; height:2.509ex;" alt="{\displaystyle i,j\in I}"></span> 및 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X_{i}\cap X_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X_{i}\cap X_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f51ed4fdbc27dccf962f4f0b2f0c5cff82f0f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.311ex; height:2.843ex;" alt="{\displaystyle x\in X_{i}\cap X_{j}}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}(x)=f_{j}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}(x)=f_{j}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5c869958dfdd9e7040308e73f8e06545ea7740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.364ex; height:3.009ex;" alt="{\displaystyle f_{i}(x)=f_{j}(x)}"></span></li></ul> <p>그렇다면 함수들을 정의역의 <a href="/wiki/%ED%95%A9%EC%A7%91%ED%95%A9" title="합집합">합집합</a>에서 공역의 합집합으로 가는 하나의 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \bigcup _{i\in I}X_{i}\to \bigcup _{i\in I}Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \bigcup _{i\in I}X_{i}\to \bigcup _{i\in I}Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3498cc352ddacf260ad4b1be84fcc73ac1de9637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.738ex; height:5.676ex;" alt="{\displaystyle f\colon \bigcup _{i\in I}X_{i}\to \bigcup _{i\in I}Y_{i}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\upharpoonright X_{i}=f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↾</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\upharpoonright X_{i}=f_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea9b2571c86a1ff758a8a5efa9bdda1a13ddd09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.3ex; height:2.509ex;" alt="{\displaystyle f\upharpoonright X_{i}=f_{i}}"></span></dd></dl> <p>로 합칠 수 있다.<sup id="cite_ref-daimm_3-0" class="reference"><a href="#cite_note-daimm-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:16, 21</sup></span> </p> <div class="mw-heading mw-heading3"><h3 id="점별_연산"><span id=".EC.A0.90.EB.B3.84_.EC.97.B0.EC.82.B0"></span>점별 연산</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=15" title="부분 편집: 점별 연산"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%A0%90%EB%B3%84_%EC%97%B0%EC%82%B0" class="mw-redirect" title="점별 연산">점별 연산</a>입니다.</div> <p>정의역이 같은 함수족 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i}\colon X\to Y_{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i}\colon X\to Y_{i})_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0fbab436a447a2ff3a08b2ce93f0307e3c52ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.251ex; height:2.843ex;" alt="{\displaystyle (f_{i}\colon X\to Y_{i})_{i\in I}}"></span>에 대하여, 공역의 <a href="/wiki/%EA%B3%B1%EC%A7%91%ED%95%A9" title="곱집합">곱집합</a>으로 가는 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i})_{i\in I}\colon X\to \prod _{i\in I}Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i})_{i\in I}\colon X\to \prod _{i\in I}Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc814b7430a99699cd3d1a6b08064f28a5086537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.608ex; height:5.676ex;" alt="{\displaystyle (f_{i})_{i\in I}\colon X\to \prod _{i\in I}Y_{i}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i})_{i\in I}\colon x\mapsto (f_{i}(x))_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{i})_{i\in I}\colon x\mapsto (f_{i}(x))_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8369a84887b12c08dffa445e90883bb2054ec09f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.062ex; height:2.843ex;" alt="{\displaystyle (f_{i})_{i\in I}\colon x\mapsto (f_{i}(x))_{i\in I}}"></span></dd></dl> <p>를 정의할 수 있다. </p><p>특수한 공역을 갖는 함수에 대하여 점별 연산을 정의할 수 있으며, 이는 위 함수와 공역 위 연산의 합성을 통해 나타낼 수 있다. 예를 들어, 두 실숫값 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b688ef73b2401ef61c4466b7b62f96eefaeebc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.735ex; height:2.509ex;" alt="{\displaystyle f,g\colon X\to \mathbb {R} }"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,g)\colon X\to \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,g)\colon X\to \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f43778e752eb5017360f533a78b0766e8df1d83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.598ex; height:3.176ex;" alt="{\displaystyle (f,g)\colon X\to \mathbb {R} ^{2}}"></span>와 실수의 덧셈의 <a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%ED%95%A9%EC%84%B1" title="함수의 합성">합성</a>을 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>와 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>의 <b>점별합</b>(點別合, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">pointwise sum</span>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d94a24abd865f6f9fd67a7df7e531cae1c769b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}"></span>라고 하며, 실수의 곱셈과의 합성을 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>와 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>의 <b>점별곱</b>(點別-, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">pointwise product</span>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bac4638bb56f14688118ce88c188c7a021eb29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}"></span>라고 한다. 구체적으로 이들은 각각 다음과 같다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g\colon X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+g\colon X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c41b6c974a5a435552e6202e646505c7005bf79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.541ex; height:2.509ex;" alt="{\displaystyle f+g\colon X\to \mathbb {R} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g\colon x\mapsto f(x)+g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+g\colon x\mapsto f(x)+g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47ea486eb59577815a430ed12dbcf1206ab5e45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.726ex; height:2.843ex;" alt="{\displaystyle f+g\colon x\mapsto f(x)+g(x)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg\colon X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg\colon X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51782469320113ec55c6891483a33a24ce20e473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.701ex; height:2.509ex;" alt="{\displaystyle fg\colon X\to \mathbb {R} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg\colon x\mapsto f(x)g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> <mo>:</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg\colon x\mapsto f(x)g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42d472367b749539e0754641040d8d5c8d7d3620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.045ex; height:2.843ex;" alt="{\displaystyle fg\colon x\mapsto f(x)g(x)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="역사"><span id=".EC.97.AD.EC.82.AC"></span>역사</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=16" title="부분 편집: 역사"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%82%BC%EA%B0%81%ED%95%A8%EC%88%98" class="mw-redirect" title="삼각함수">삼각함수</a>와 같은 특정 함수에 대한 연구는 오래전부터 있어 왔다. 16세기 <a href="/wiki/%EB%9D%BC%EC%9D%B4%ED%94%84%EC%B9%98%ED%9E%88_%EB%8C%80%ED%95%99%EA%B5%90" title="라이프치히 대학교">라이프치히 대학교</a>의 수학 교수이자 <a href="/wiki/%EC%BD%94%ED%8E%98%EB%A5%B4%EB%8B%88%EC%BF%A0%EC%8A%A4" class="mw-redirect" title="코페르니쿠스">코페르니쿠스</a>의 《<a href="/wiki/%EC%B2%9C%EA%B5%AC%EC%9D%98_%ED%9A%8C%EC%A0%84%EC%97%90_%EA%B4%80%ED%95%98%EC%97%AC" title="천구의 회전에 관하여">천구의 회전에 관하여</a>》가 출간되는데 큰 역할을 하였던 <a href="/wiki/%EB%A0%88%ED%8B%B0%EC%BF%A0%EC%8A%A4" class="mw-redirect" title="레티쿠스">레티쿠스</a>는 1596년 《팔라티누스 삼각형 서(書)》(<span style="font-size: smaller;"><a href="/wiki/%EB%9D%BC%ED%8B%B4%EC%96%B4" title="라틴어">라틴어</a>: </span><span lang="la">Opus Palatinum de triangulis</span>)에서 삼각함수표를 정리하여 발표하기도 하였다.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> 그러나 당시의 연구는 현재의 함수 정의에 확립되어 있는 <a href="/wiki/%EA%B4%80%EA%B3%84_(%EC%88%98%ED%95%99)" title="관계 (수학)">관계</a>에 대한 개념이 없이 단순히 계산의 편의를 도모하기 위한 것이었다. 한편, <a href="/wiki/%EB%A5%B4%EB%84%A4_%EB%8D%B0%EC%B9%B4%EB%A5%B4%ED%8A%B8" title="르네 데카르트">르네 데카르트</a>는 <a href="/wiki/%EB%8D%B0%EC%B9%B4%EB%A5%B4%ED%8A%B8_%EC%A2%8C%ED%91%9C%EA%B3%84" title="데카르트 좌표계">데카르트 좌표계</a>를 이용하여 오늘날 함수의 관계식에 해당하는 <a href="/wiki/%EB%B0%A9%EC%A0%95%EC%8B%9D" title="방정식">방정식</a>을 <a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%EA%B7%B8%EB%9E%98%ED%94%84" title="함수의 그래프">그래프</a>로 표현하는 방법을 제시하였다.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>17세기에 도입한 대부분의 함수는 함수 개념이 충분히 인식되기 이전에는 곡선, 특히 운동 궤적으로서 연구되었다. 1667년, <a href="/wiki/%EC%A0%9C%EC%9E%84%EC%8A%A4_%EA%B7%B8%EB%A0%88%EA%B3%A0%EB%A6%AC" title="제임스 그레고리">제임스 그레고리</a>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">James Gregory</span>)는 논문 《원과 쌍곡선의 구적법에 대하여》(<span style="font-size: smaller;"><a href="/wiki/%EB%9D%BC%ED%8B%B4%EC%96%B4" title="라틴어">라틴어</a>: </span><span lang="la">Vera Circuli et Hyperbolae Quadratura</span>)에서 함수를 다른 양들에 대한 대수 연산 및 극한 연산을 통해 얻는 양으로 정의하였다. 1665년부터, 아이작 뉴턴은 줄곧 “플루언트”(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">fluent</span>)라는 용어로 변수 간 관계를 지칭하였다. 1673년, <a href="/wiki/%EA%B3%A0%ED%8A%B8%ED%94%84%EB%A6%AC%ED%8A%B8_%EB%B9%8C%ED%97%AC%EB%A6%84_%EB%9D%BC%EC%9D%B4%ED%94%84%EB%8B%88%EC%B8%A0" title="고트프리트 빌헬름 라이프니츠">고트프리트 빌헬름 라이프니츠</a>는 오늘날 쓰이는 용어인 “함수”(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">function</span>)을 곡선 위 점에 따라 변화하는 양으로 정의하였다. 1697년, <a href="/wiki/%EC%9A%94%ED%95%9C_%EB%B2%A0%EB%A5%B4%EB%88%84%EC%9D%B4" title="요한 베르누이">요한 베르누이</a>는 함수를 상수와 변수가 대수 연산 및 초월 연산을 통해 구성하는 양으로 정의하였으며, 1698년에 라이프니츠의 용어를 채택하였다. 1714년, 라이프니츠는 저서 《역사》(<span style="font-size: smaller;"><a href="/wiki/%EB%9D%BC%ED%8B%B4%EC%96%B4" title="라틴어">라틴어</a>: </span><span lang="la">historia</span>)에서 함수를 변수에 의존하는 양으로 정의하였다. 그러나, 그는 여태 <a href="/wiki/%EB%AF%B8%EB%B6%84_%EA%B0%80%EB%8A%A5%ED%95%9C_%ED%95%A8%EC%88%98" class="mw-redirect" title="미분 가능한 함수">미분 가능한 함수</a>만을 다루었다. </p><p><a href="/wiki/%EB%A0%88%EC%98%A8%ED%95%98%EB%A5%B4%ED%8A%B8_%EC%98%A4%EC%9D%BC%EB%9F%AC" title="레온하르트 오일러">레온하르트 오일러</a>는 1734년에 오늘날 쓰이는 표기법 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span>를 도입하였다. 또한, 오일러는 1748년에 저서 《무한 해석 입문》(<span style="font-size: smaller;"><a href="/wiki/%EB%9D%BC%ED%8B%B4%EC%96%B4" title="라틴어">라틴어</a>: </span><span lang="la">Introductio in Analysin Infinitorum</span>)에서 함수를 변수와 상수로 구성된 임의의 해석적 수식으로 정의하였으며, 1775년에 저서 《미분학 입문》(<span style="font-size: smaller;"><a href="/wiki/%EB%9D%BC%ED%8B%B4%EC%96%B4" title="라틴어">라틴어</a>: </span><span lang="la">Institutiones Calculi Differentialis</span>)에서 변수에 의존하며 그 변화에 따라 변화하는 또 다른 변수로 정의하였다. </p><p>1797년, <a href="/w/index.php?title=%EC%8B%A4%EB%B2%A0%EC%8A%A4%ED%8A%B8%EB%A5%B4_%ED%94%84%EB%9E%91%EC%88%98%EC%95%84_%EB%9D%BC%ED%81%AC%EB%A3%A8%EC%95%84&action=edit&redlink=1" class="new" title="실베스트르 프랑수아 라크루아 (없는 문서)">실베스트르 프랑수아 라크루아</a>(<span style="font-size: smaller;"><a href="/wiki/%ED%94%84%EB%9E%91%EC%8A%A4%EC%96%B4" title="프랑스어">프랑스어</a>: </span><span lang="fr">Sylvestre-François Lacroix</span>)는 저서 《미분과 적분에 대하여》(<span style="font-size: smaller;"><a href="/wiki/%ED%94%84%EB%9E%91%EC%8A%A4%EC%96%B4" title="프랑스어">프랑스어</a>: </span><span lang="fr">Traité du Calcul Différentiel et du Calcul Intégral</span>)에서 수식으로 표현될 필요가 없는, 더 넓은 함수의 개념을 도입하였으며, 5차 방정식의 근이 5차 방정식의 계수의 함수라는 예시를 들었다. 1811~15년, <a href="/wiki/%EC%A1%B0%EC%A0%9C%ED%94%84%EB%A3%A8%EC%9D%B4_%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A3%BC" title="조제프루이 라그랑주">조제프루이 라그랑주</a>는 저서 《역학 해석》(<span style="font-size: smaller;"><a href="/wiki/%EB%9D%BC%ED%8B%B4%EC%96%B4" title="라틴어">라틴어</a>: </span><span lang="la">Mecanique analytique</span>)에서 “함수”라는 용어를 거의 모든 유형의 함수에서 사용하였다. </p><p><a href="/wiki/%EC%A1%B0%EC%A0%9C%ED%94%84_%ED%91%B8%EB%A6%AC%EC%97%90" title="조제프 푸리에">조제프 푸리에</a>는 함수가 해석적 수식으로 표현될 수 있을 필요가 없다고 주장하였으나, 동시에 모든 함수는 푸리에 급수로 표현될 수 있다고 주장하였다. 그러나 그는 임의의 유한 구간에서 유한 개의 불연속점만을 갖는 함수만을 다루었다. </p><p>1837년, <a href="/wiki/%ED%8E%98%ED%84%B0_%EA%B5%AC%EC%8A%A4%ED%83%80%ED%94%84_%EB%A5%B4%EC%A3%88_%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88" title="페터 구스타프 르죈 디리클레">페터 구스타프 르죈 디리클레</a>는 논문 《완전히 임의인 함수의 사인 및 코사인 함수 표현에 대하여》(<span style="font-size: smaller;"><a href="/wiki/%EB%8F%85%EC%9D%BC%EC%96%B4" title="독일어">독일어</a>: </span><span lang="de">Ober die Darstellung ganz willkurlicher Functionen durch Sinus-und Cosinusreihen</span>)에서, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>가 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>의 함수라는 것을 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>의 주어진 구간에서의 임의의 값에 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>의 유일한 값이 대응하는 것으로 정의하였으며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>가 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>에 따라 어떤 법칙을 통해 결정되거나, 수학 공식으로 표현될 필요는 없다고 설명하였다. 이는 오늘날에도 사용되는 정의이다. </p><p>함수의 현대적 정의는 <a href="/wiki/%EA%B2%8C%EC%98%A4%EB%A5%B4%ED%81%AC_%EC%B9%B8%ED%86%A0%EC%96%B4" title="게오르크 칸토어">게오르크 칸토어</a>가 제기한 <a href="/wiki/%EC%A7%91%ED%95%A9%EB%A1%A0" title="집합론">집합론</a>에 기반한 것이다. <a href="/wiki/%EB%B2%84%ED%8A%B8%EB%9F%B0%EB%93%9C_%EB%9F%AC%EC%85%80" title="버트런드 러셀">버트런드 러셀</a>은 <a href="/wiki/%EC%A7%91%ED%95%A9" title="집합">집합</a>을 기반으로 수학의 공리를 재서술하면서 함수 역시 이를 기반으로 재정의하였다.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="어원"><span id=".EC.96.B4.EC.9B.90"></span>어원</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=17" title="부분 편집: 어원"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>17세기 <a href="/wiki/%EA%B3%A0%ED%8A%B8%ED%94%84%EB%A6%AC%ED%8A%B8_%EB%B9%8C%ED%97%AC%EB%A6%84_%EB%9D%BC%EC%9D%B4%ED%94%84%EB%8B%88%EC%B8%A0" title="고트프리트 빌헬름 라이프니츠">고트프리트 빌헬름 라이프니츠</a>는 수학 저서에서 라틴어 단어 <span lang="la">functio</span>를 주로 ‘기능’이란 뜻으로 썼다. 이후 <a href="/wiki/%EC%9A%94%ED%95%9C_%EB%B2%A0%EB%A5%B4%EB%88%84%EC%9D%B4" title="요한 베르누이">요한 베르누이</a> 등이 <span lang="la">functio</span>를 기술적인 해석학 용어로 쓰기 시작했다. 이것이 다른 유럽 언어로 전파되었다. </p><p>‘함수(函數)’라는 용어를 쓰기 시작한 사람은 <a href="/wiki/%EC%9D%B4%EC%84%A0%EB%9E%80" title="이선란">이선란</a>과 <span title="대상 항목은 아직 없습니다. 신규 작성, 또는 다른 언어로부터의 번역이 필요합니다.]"><a href="/w/index.php?title=%EC%95%8C%EB%A0%89%EC%82%B0%EB%8D%94_%EC%99%80%EC%9D%BC%EB%A6%AC&action=edit&redlink=1" class="new" title="알렉산더 와일리 (없는 문서)">알렉산더 와일리</a><span style="font-size: 0.77em; font-weight: normal;" class="noprint">(<a href="https://en.wikipedia.org/wiki/Alexander_Wylie_(missionary)" class="extiw" title="en:Alexander Wylie (missionary)">영어판</a>)</span></span>이다. 그들은 번역서 《대수학(代數學)》(1859)과 《대미적습급(代微積拾級)》(1859)에서 영어 <span lang="en">function</span>의 번역어로 ‘함수(函數)’라는 단어를 썼다. 드모르간은 《The Elements of Algebra》에서 function을 ‘변수를 담고 있는 식’으로 소개하는데, 이를 ‘상자’·‘담다’라는 뜻을 가진 한자 함(函)을 써서 의역한 것이다. </p> <blockquote class="templatequote"> <p>Any expression which contains x in any way is called a function of x: thus a+x, a+bx<sup>2</sup>, &c.<br /> x를 어떤 형태로든 담고 있는 모든 식은 x의 함수로 부른다: 즉 a+x, a+bx<sup>2</sup> 등이다. </p> <div class="templatequotecite"><cite>— <a href="/wiki/%EC%98%A4%EA%B1%B0%EC%8A%A4%ED%84%B0%EC%8A%A4_%EB%93%9C%EB%AA%A8%EB%A5%B4%EA%B0%84" class="mw-redirect" title="오거스터스 드모르간">오거스터스 드모르간</a>, 《The Elements of Algebra》<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></cite></div> </blockquote> <blockquote class="templatequote"> <p>凡此變數中函彼變數者，則此為彼之函數。<br /> 이 변수가 안에 그 변수를 포함한다면, 이 (변수)는 그 (변수)의 함수라 한다. </p> <div class="templatequotecite"><cite>— 《대수학(代數學)》</cite></div> </blockquote> <p>‘함수’가 영어 단어 <span lang="en">function</span>의 발음을 음역한 단어라는 설이 있지만 두 발음이 크게 달라 근거가 희박하다. 또한 이선란이 만든 다른 번역어 <a href="/wiki/%EC%83%81%EC%88%98" title="상수">상수</a>·<a href="/wiki/%EB%B3%80%EC%88%98_(%EC%88%98%ED%95%99)" title="변수 (수학)">변수</a>·<a href="/wiki/%EA%B3%84%EC%88%98" title="계수">계수</a>·<a href="/wiki/%EC%A7%80%EC%88%98" class="mw-redirect" title="지수">지수</a>·<a href="/wiki/%EA%B8%89%EC%88%98_(%EC%88%98%ED%95%99)" title="급수 (수학)">급수</a> 중의 그 어떤 것도 음역이 아니다. </p> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=18" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-이상구-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-이상구_1-0">가</a></sup> <sup><a href="#cite_ref-이상구_1-1">나</a></sup> <sup><a href="#cite_ref-이상구_1-2">다</a></sup> <sup><a href="#cite_ref-이상구_1-3">라</a></sup> <sup><a href="#cite_ref-이상구_1-4">마</a></sup></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://matrix.skku.ac.kr/sglee/calculus/new1.pdf">1. 기본개념</a>, 성균관대학교 대수학 연구실 이상구 교수 홈페이지</span> </li> <li id="cite_note-박은순-2"><span class="mw-cite-backlink"><a href="#cite_ref-박은순_2-0">↑</a></span> <span class="reference-text"><cite class="citation book">박은순 (2008). 《쉬운 미분·적분학》. 숭실대학교출판부. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/89-7450-235-6" title="특수:책찾기/89-7450-235-6"><bdi>89-7450-235-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%EC%89%AC%EC%9A%B4+%EB%AF%B8%EB%B6%84%C2%B7%EC%A0%81%EB%B6%84%ED%95%99&rft.pub=%EC%88%AD%EC%8B%A4%EB%8C%80%ED%95%99%EA%B5%90%EC%B6%9C%ED%8C%90%EB%B6%80&rft.date=2008&rft.isbn=89-7450-235-6&rft.au=%EB%B0%95%EC%9D%80%EC%88%9C&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%95%A8%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-daimm-3"><span class="mw-cite-backlink"><a href="#cite_ref-daimm_3-0">↑</a></span> <span class="reference-text"><cite class="citation book">戴牧民; 陈海燕; 郑顶伟 (2011). 《公理集合论导引》 (중국어). 北京: 科学出版社. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-7-03-031276-1" title="특수:책찾기/978-7-03-031276-1"><bdi>978-7-03-031276-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%E5%85%AC%E7%90%86%E9%9B%86%E5%90%88%E8%AE%BA%E5%AF%BC%E5%BC%95&rft.place=%E5%8C%97%E4%BA%AC&rft.pub=%E7%A7%91%E5%AD%A6%E5%87%BA%E7%89%88%E7%A4%BE&rft.date=2011&rft.isbn=978-7-03-031276-1&rft.au=%E6%88%B4%E7%89%A7%E6%B0%91&rft.au=%E9%99%88%E6%B5%B7%E7%87%95&rft.au=%E9%83%91%E9%A1%B6%E4%BC%9F&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%95%A8%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">과학동아편집실, 수학자를 알면 공식이 보인다, 성우, 2002, 72-74쪽</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">과학동아편집실, 수학자를 알면 공식이 보인다, 성우, 2002, 82-88쪽</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://fair-use.org/bertrand-russell/the-principles-of-mathematics/index">The Principles of Mathematics</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><cite class="citation book">De Morgan, Augustus (1837). <a rel="nofollow" class="external text" href="https://books.google.co.kr/books?id=Q49aAAAAcAAJ&pg=PA168&lpg=PA168&dq=%22Any+expression+which+contains+x+in+any+way+is+called+a+function+of+x%22&source=bl&ots=-qqUzHuiEs&sig=ACfU3U2cnqcjfZGm1XiVffDr0LTUhCuf4g&hl=en&sa=X&ved=2ahUKEwjovPC97p7sAhWb7WEKHfM8CnEQ6AEwAHoECAEQAg#v=onepage&q=%22Any%20expression%20which%20contains%20x%20in%20any%20way%20is%20called%20a%20function%20of%20x%22&f=false">《The Elements of Algebra》</a>. 168쪽<span class="reference-accessdate">. 2020년 10월 6일에 확인함</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Elements+of+Algebra&rft.pages=168&rft.date=1837&rft.aulast=De+Morgan&rft.aufirst=Augustus&rft_id=https%3A%2F%2Fbooks.google.co.kr%2Fbooks%3Fid%3DQ49aAAAAcAAJ%26pg%3DPA168%26lpg%3DPA168%26dq%3D%2522Any%2Bexpression%2Bwhich%2Bcontains%2Bx%2Bin%2Bany%2Bway%2Bis%2Bcalled%2Ba%2Bfunction%2Bof%2Bx%2522%26source%3Dbl%26ots%3D-qqUzHuiEs%26sig%3DACfU3U2cnqcjfZGm1XiVffDr0LTUhCuf4g%26hl%3Den%26sa%3DX%26ved%3D2ahUKEwjovPC97p7sAhWb7WEKHfM8CnEQ6AEwAHoECAEQAg%23v%3Donepage%26q%3D%2522Any%2520expression%2520which%2520contains%2520x%2520in%2520any%2520way%2520is%2520called%2520a%2520function%2520of%2520x%2522%26f%3Dfalse&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%95%A8%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%95%A8%EC%88%98&action=edit&section=19" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r36480479">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box metadata side-box-right plainlinks"><style data-mw-deduplicate="TemplateStyles:r36480595">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist"><b><a href="/wiki/%EC%9C%84%ED%82%A4%EB%AF%B8%EB%94%94%EC%96%B4_%EA%B3%B5%EC%9A%A9" title="위키미디어 공용">위키미디어 공용</a></b>에 관련된<br />미디어 분류가 있습니다.<div style="padding-left:1em;"><b><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Functions?uselang=ko">함수</a></b></div></div></div> </div> <ul><li><cite class="citation web">이철희. <a rel="nofollow" class="external text" href="http://wiki.mathnt.net/index.php?title=%ED%95%A8%EC%88%98">“함수”</a>. 《수학노트》.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=%EC%88%98%ED%95%99%EB%85%B8%ED%8A%B8&rft.atitle=%ED%95%A8%EC%88%98&rft.au=%EC%9D%B4%EC%B2%A0%ED%9D%AC&rft_id=http%3A%2F%2Fwiki.mathnt.net%2Findex.php%3Ftitle%3D%25ED%2595%25A8%25EC%2588%2598&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%95%A8%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Function">“Function”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Function&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FFunction&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%95%A8%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Function.html">“Function”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Function&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FFunction.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%95%A8%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://ncatlab.org/nlab/show/function">“Function”</a>. 《nLab》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Function&rft_id=http%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Ffunction&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%95%A8%EC%88%98" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.mw-parser-output .hlist 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class="navbox-group" style="width:1%"><a href="/wiki/%EA%B8%B0%EC%88%98_(%EC%88%98%ED%95%99)" title="기수 (수학)">기수</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%A7%91%ED%95%A9%EC%9D%98_%ED%81%AC%EA%B8%B0" title="집합의 크기">집합의 크기</a></li> <li><a href="/wiki/%EC%B9%B8%ED%86%A0%EC%96%B4%EC%9D%98_%EC%A0%95%EB%A6%AC" title="칸토어의 정리">칸토어의 정리</a> <ul><li><a href="/wiki/%EB%8C%80%EA%B0%81%EC%84%A0_%EB%85%BC%EB%B2%95" title="대각선 논법">대각선 논법</a></li></ul></li> <li><a href="/wiki/%EC%95%8C%EB%A0%88%ED%94%84_%EC%88%98" title="알레프 수">알레프 수</a></li> <li><a href="/wiki/%EB%B2%A0%ED%8A%B8_%EC%88%98" title="베트 수">베트 수</a></li> <li><a href="/wiki/%EA%B7%B9%ED%95%9C_%EA%B8%B0%EC%88%98" title="극한 기수">극한 기수</a></li> <li><a href="/wiki/%EA%B8%B0%EB%A9%9C_%ED%95%A8%EC%88%98" title="기멜 함수">기멜 함수</a></li> <li><a href="/wiki/%EA%B0%80%EB%8A%A5_%EA%B3%B5%EC%A2%85%EB%8F%84" title="가능 공종도">가능 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style="width:1%">역설의 해소</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%86%8C%EB%B0%95%ED%95%9C_%EC%A7%91%ED%95%A9%EB%A1%A0" title="소박한 집합론">소박한 집합론</a></li> <li><a href="/wiki/%EB%9F%AC%EC%85%80%EC%9D%98_%EC%97%AD%EC%84%A4" title="러셀의 역설">러셀의 역설</a></li> <li><a href="/wiki/%EC%B9%B8%ED%86%A0%EC%96%B4_%EC%97%AD%EC%84%A4" title="칸토어 역설">칸토어 역설</a></li> <li><a href="/wiki/%EB%B6%80%EB%9E%84%EB%A6%AC%ED%8F%AC%EB%A5%B4%ED%8B%B0_%EC%97%AD%EC%84%A4" title="부랄리포르티 역설">부랄리포르티 역설</a></li> <li><a href="/wiki/%EB%AA%A8%EC%9E%84_(%EC%A7%91%ED%95%A9%EB%A1%A0)" title="모임 (집합론)">모임</a></li> <li><a href="/wiki/%EC%9C%A0%ED%98%95_%EC%9D%B4%EB%A1%A0" title="유형 이론">유형 이론</a> <ul><li>《<a href="/wiki/%EC%88%98%ED%95%99_%EC%9B%90%EB%A6%AC" title="수학 원리">수학 원리</a>》</li></ul></li> <li><a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">체르멜로-프렝켈 집합론</a></li> <li><a href="/wiki/%EC%83%88_%EA%B8%B0%EC%B4%88" title="새 기초">새 기초</a></li> <li><a href="/wiki/%ED%8F%B0_%EB%85%B8%EC%9D%B4%EB%A7%8C-%EB%B2%A0%EB%A5%B4%EB%82%98%EC%9D%B4%EC%8A%A4-%EA%B4%B4%EB%8D%B8_%EC%A7%91%ED%95%A9%EB%A1%A0" title="폰 노이만-베르나이스-괴델 집합론">폰 노이만-베르나이스-괴델 집합론</a></li> <li><a href="/w/index.php?title=%EB%AA%A8%EC%8A%A4-%EC%BC%88%EB%A6%AC_%EC%A7%91%ED%95%A9%EB%A1%A0&action=edit&redlink=1" class="new" title="모스-켈리 집합론 (없는 문서)">모스-켈리 집합론</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" title="선택 공리">선택 공리</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EA%B0%80%EC%82%B0_%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" class="mw-redirect" title="가산 선택 공리">가산 선택 공리</a></li> <li><a href="/wiki/%EC%9D%98%EC%A1%B4%EC%A0%81_%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" class="mw-redirect" title="의존적 선택 공리">의존적 선택 공리</a></li> <li><a href="/wiki/%EC%B4%88%EB%A5%B8_%EB%B3%B4%EC%A1%B0%EC%A0%95%EB%A6%AC" title="초른 보조정리">초른 보조정리</a></li> <li><a href="/wiki/%EC%8A%88%ED%95%84%EB%9D%BC%EC%9D%B8_%ED%99%95%EC%9E%A5%EC%A0%95%EB%A6%AC" title="슈필라인 확장정리">슈필라인 확장정리</a></li> <li><a href="/wiki/%EB%B0%94%EB%82%98%ED%9D%90-%ED%83%80%EB%A5%B4%EC%8A%A4%ED%82%A4_%EC%97%AD%EC%84%A4" title="바나흐-타르스키 역설">바나흐-타르스키 역설</a></li> <li><a href="/wiki/%ED%8B%B0%ED%98%B8%EB%85%B8%ED%94%84_%EC%A0%95%EB%A6%AC" title="티호노프 정리">티호노프 정리</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">집합론의 <a href="/wiki/%EB%AA%A8%ED%98%95_%EC%9D%B4%EB%A1%A0" title="모형 이론">모형</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EA%B5%AC%EC%84%B1_%EA%B0%80%EB%8A%A5_%EC%A0%84%EC%B2%B4" title="구성 가능 전체">구성 가능 전체</a></li> <li><a href="/wiki/%EC%B6%94%EC%9D%B4%EC%A0%81_%EC%A7%91%ED%95%A9" title="추이적 집합">추이적 집합</a></li> <li><a href="/wiki/%EC%B6%94%EC%9D%B4%EC%A0%81_%EB%AA%A8%ED%98%95" title="추이적 모형">추이적 모형</a></li> <li><a href="/wiki/%EC%88%9C%EC%84%9C%EC%88%98_%EC%A0%95%EC%9D%98_%EA%B0%80%EB%8A%A5_%EC%A7%91%ED%95%A9" title="순서수 정의 가능 집합">순서수 정의 가능 집합</a></li> <li><a href="/wiki/%EA%B0%95%EC%A0%9C%EB%B2%95" title="강제법">강제법</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%ED%81%B0_%EA%B8%B0%EC%88%98" title="큰 기수">큰 기수</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EB%8F%84%EB%8B%AC_%EB%B6%88%EA%B0%80%EB%8A%A5%ED%95%9C_%EA%B8%B0%EC%88%98" title="도달 불가능한 기수">도달 불가능한 기수</a></li> <li><a href="/wiki/%EA%B0%80%EC%B8%A1_%EA%B8%B0%EC%88%98" title="가측 기수">가측 기수</a></li> <li><a href="/wiki/%EC%95%BD%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B8%B0%EC%88%98" title="약콤팩트 기수">약콤팩트 기수</a></li> <li><a href="/wiki/%EA%B0%95%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B8%B0%EC%88%98" title="강콤팩트 기수">강콤팩트 기수</a></li> <li><a href="/wiki/%EC%B4%88%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B8%B0%EC%88%98" title="초콤팩트 기수">초콤팩트 기수</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">ZF</a>와 독립된 명제</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EC%97%B0%EC%86%8D%EC%B2%B4_%EA%B0%80%EC%84%A4" title="연속체 가설">연속체 가설</a></li> <li><a href="/wiki/%ED%8A%B9%EC%9D%B4_%EA%B8%B0%EC%88%98_%EA%B0%80%EC%84%A4" title="특이 기수 가설">특이 기수 가설</a></li> <li><a href="/wiki/%EC%88%98%EC%8A%AC%EB%A6%B0_%EA%B0%80%EC%84%A4" class="mw-redirect" title="수슬린 가설">수슬린 가설</a></li> <li><a href="/wiki/%ED%99%94%EC%9D%B4%ED%8A%B8%ED%97%A4%EB%93%9C_%EB%AC%B8%EC%A0%9C" title="화이트헤드 문제">화이트헤드 문제</a></li> <li><a href="/wiki/%EA%B2%B0%EC%A0%95_%EC%A7%91%ED%95%A9" title="결정 집합">결정 공리</a></li> <li><a href="/wiki/%EB%A7%88%ED%8B%B4_%EA%B3%B5%EB%A6%AC" title="마틴 공리">마틴 공리</a></li> <li><a href="/wiki/%EB%8B%A4%EC%9D%B4%EC%95%84%EB%AA%AC%EB%93%9C_%EC%9B%90%EB%A6%AC" class="mw-redirect" title="다이아몬드 원리">다이아몬드 원리</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36429174"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%A0%84%EA%B1%B0_%ED%86%B5%EC%A0%9C" title="위키백과:전거 통제">전거 통제</a>: 국가 <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11348#identifiers" title="위키데이터에서 편집하기"><img alt="위키데이터에서 편집하기" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="uid"><abbr title="Fonctions (mathématiques)"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11946892t">프랑스</a></abbr></span></li> <li><span class="uid"><abbr title="Fonctions (mathématiques)"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11946892t">BnF 데이터</a></abbr></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4071510-3">독일</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="http://uli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007553160705171">이스라엘</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85052327">미국</a></span></li> <li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00564960">일본</a></span></li> <li><span class="uid"><abbr title="funkce"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph114594&CON_LNG=ENG">체코</a></abbr></span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">원본 주소 "<a dir="ltr" href="https://ko.wikipedia.org/w/index.php?title=함수&oldid=37151393">https://ko.wikipedia.org/w/index.php?title=함수&oldid=37151393</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%B6%84%EB%A5%98" title="특수:분류">분류</a>: <ul><li><a href="/wiki/%EB%B6%84%EB%A5%98:%ED%95%A8%EC%88%98%EC%99%80_%EC%82%AC%EC%83%81" title="분류:함수와 사상">함수와 사상</a></li><li><a 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