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href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%A1%D1%8A%D0%B7%D0%B4%D0%B0%D0%B2%D0%B0%D0%BD%D0%B5_%D0%BD%D0%B0_%D1%81%D0%BC%D0%B5%D1%82%D0%BA%D0%B0&returnto=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Насърчаваме Ви да си създадете сметка и да влезете, въпреки че не е задължително." class=""><span>Създаване на сметка</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=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%A0%D0%B5%D0%B3%D0%B8%D1%81%D1%82%D1%80%D0%B8%D1%80%D0%B0%D0%BD%D0%B5_%D0%B8%D0%BB%D0%B8_%D0%B2%D0%BB%D0%B8%D0%B7%D0%B0%D0%BD%D0%B5&returnto=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Препоръчваме Ви да влезете, въпреки, че не е задължително. 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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Влизане</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"> Страници за излезли от системата редактори <a href="/wiki/%D0%9F%D0%BE%D0%BC%D0%BE%D1%89:%D0%92%D1%8A%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" aria-label="Научете повече за редактирането"><span>научете повече</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/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%9C%D0%BE%D0%B8%D1%82%D0%B5_%D0%BF%D1%80%D0%B8%D0%BD%D0%BE%D1%81%D0%B8" title="Списък на промените, направени от този IP адрес [y]" accesskey="y"><span>Приноси</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%9C%D0%BE%D1%8F%D1%82%D0%B0_%D0%B1%D0%B5%D1%81%D0%B5%D0%B4%D0%B0" title="Дискусия относно редакциите от този адрес [n]" accesskey="n"><span>Беседа</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="Сайт"> <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="Съдържание" 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">Съдържание</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">преместване към страничната лента</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">скриване</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">Начало</div> </a> </li> <li id="toc-Терминология" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Терминология"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Терминология</span> </div> </a> <ul id="toc-Терминология-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Формални_дефиниции" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Формални_дефиниции"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Формални дефиниции</span> </div> </a> <button aria-controls="toc-Формални_дефиниции-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>Превключване на подраздел Формални дефиниции</span> </button> <ul id="toc-Формални_дефиниции-sublist" class="vector-toc-list"> <li id="toc-Функция-оригинал" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Функция-оригинал"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Функция-оригинал</span> </div> </a> <ul id="toc-Функция-оригинал-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-История_на_понятието" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#История_на_понятието"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>История на понятието</span> </div> </a> <ul id="toc-История_на_понятието-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Видове_функции" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Видове_функции"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Видове функции</span> </div> </a> <ul id="toc-Видове_функции-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Свойства_на_функциите" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Свойства_на_функциите"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Свойства на функциите</span> </div> </a> <button aria-controls="toc-Свойства_на_функциите-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>Превключване на подраздел Свойства на функциите</span> </button> <ul id="toc-Свойства_на_функциите-sublist" class="vector-toc-list"> <li id="toc-Дефинируемост" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Дефинируемост"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Дефинируемост</span> </div> </a> <ul id="toc-Дефинируемост-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Диференцируемост" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Диференцируемост"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Диференцируемост</span> </div> </a> <ul id="toc-Диференцируемост-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Непрекъснатост" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Непрекъснатост"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Непрекъснатост</span> </div> </a> <ul id="toc-Непрекъснатост-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Интегруемост" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Интегруемост"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Интегруемост</span> </div> </a> <ul id="toc-Интегруемост-sublist" class="vector-toc-list"> <li id="toc-Интегруемост_в_интервал" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Интегруемост_в_интервал"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4.1</span> <span>Интегруемост в интервал</span> </div> </a> <ul id="toc-Интегруемост_в_интервал-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Монотонност" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Монотонност"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Монотонност</span> </div> </a> <ul id="toc-Монотонност-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Графика_на_функция" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Графика_на_функция"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Графика на функция</span> </div> </a> <button aria-controls="toc-Графика_на_функция-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>Превключване на подраздел Графика на функция</span> </button> <ul id="toc-Графика_на_функция-sublist" class="vector-toc-list"> <li id="toc-Екстремуми_на_функция" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Екстремуми_на_функция"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Екстремуми на функция</span> </div> </a> <ul id="toc-Екстремуми_на_функция-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Инфлексна_точка" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Инфлексна_точка"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Инфлексна точка</span> </div> </a> <ul id="toc-Инфлексна_точка-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Вижте_също" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Вижте_също"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Вижте също</span> </div> </a> <ul id="toc-Вижте_също-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Източници" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Източници"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Източници</span> </div> </a> <ul id="toc-Източници-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="Съдържание" 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="Скриване/показване на съдържанието" > <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">Скриване/показване на съдържанието</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">Функция</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="Към статията на друг език. Налична на 120 езика" > <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">120 езика</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 – африканс" lang="af" hreflang="af" data-title="Funksie" data-language-autonym="Afrikaans" data-language-local-name="африканс" 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) – швейцарски немски" lang="gsw" hreflang="gsw" data-title="Funktion (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="швейцарски немски" 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 – амхарски" lang="am" hreflang="am" data-title="አስረካቢ" data-language-autonym="አማርኛ" data-language-local-name="амхарски" 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 – арагонски" lang="an" hreflang="an" data-title="Función matematica" data-language-autonym="Aragonés" data-language-local-name="арагонски" 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 – арабски" lang="ar" hreflang="ar" data-title="دالة" data-language-autonym="العربية" data-language-local-name="арабски" 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 – Moroccan Arabic" lang="ary" hreflang="ary" data-title="دالة" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" 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 – астурски" lang="ast" hreflang="ast" data-title="Función matemática" data-language-autonym="Asturianu" data-language-local-name="астурски" 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) – азербайджански" lang="az" hreflang="az" data-title="Funksiya (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="азербайджански" 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) – башкирски" lang="ba" hreflang="ba" data-title="Функция (математика)" data-language-autonym="Башҡортса" data-language-local-name="башкирски" 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) – беларуски" lang="be" hreflang="be" data-title="Функцыя (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="беларуски" 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-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) – бенгалски" lang="bn" hreflang="bn" data-title="অপেক্ষক (গণিত)" data-language-autonym="বাংলা" data-language-local-name="бенгалски" 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) – босненски" lang="bs" hreflang="bs" data-title="Funkcija (matematika)" data-language-autonym="Bosanski" data-language-local-name="босненски" 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 – каталонски" lang="ca" hreflang="ca" data-title="Funció" data-language-autonym="Català" data-language-local-name="каталонски" 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) – кюрдски (централен)" lang="ckb" hreflang="ckb" data-title="فانکشن (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="кюрдски (централен)" 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) – чешки" lang="cs" hreflang="cs" data-title="Funkce (matematika)" data-language-autonym="Čeština" data-language-local-name="чешки" 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) – чувашки" lang="cv" hreflang="cv" data-title="Функци (математика)" data-language-autonym="Чӑвашла" data-language-local-name="чувашки" 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 – уелски" lang="cy" hreflang="cy" data-title="Ffwythiant" data-language-autonym="Cymraeg" data-language-local-name="уелски" 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) – датски" lang="da" hreflang="da" data-title="Funktion (matematik)" data-language-autonym="Dansk" data-language-local-name="датски" 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) – немски" lang="de" hreflang="de" data-title="Funktion (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="немски" 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 – гръцки" lang="el" hreflang="el" data-title="Συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="гръцки" 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) – английски" lang="en" hreflang="en" data-title="Function (mathematics)" data-language-autonym="English" data-language-local-name="английски" 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) – есперанто" lang="eo" hreflang="eo" data-title="Funkcio (matematiko)" data-language-autonym="Esperanto" data-language-local-name="есперанто" 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) – испански" lang="es" hreflang="es" data-title="Función (matemática)" data-language-autonym="Español" data-language-local-name="испански" 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) – естонски" lang="et" hreflang="et" data-title="Funktsioon (matemaatika)" data-language-autonym="Eesti" data-language-local-name="естонски" 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) – баски" lang="eu" hreflang="eu" data-title="Funtzio (matematika)" data-language-autonym="Euskara" data-language-local-name="баски" 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 – персийски" lang="fa" hreflang="fa" data-title="تابع" data-language-autonym="فارسی" data-language-local-name="персийски" 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 – фински" lang="fi" hreflang="fi" data-title="Funktio" data-language-autonym="Suomi" data-language-local-name="фински" 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) – фиджийски" lang="fj" hreflang="fj" data-title="Cakacaka (fika)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="фиджийски" 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 – фарьорски" lang="fo" hreflang="fo" data-title="Funksjón" data-language-autonym="Føroyskt" data-language-local-name="фарьорски" 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) – френски" lang="fr" hreflang="fr" data-title="Fonction (mathématiques)" data-language-autonym="Français" data-language-local-name="френски" 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 – северен фризийски" lang="frr" hreflang="frr" data-title="Funksion" data-language-autonym="Nordfriisk" data-language-local-name="северен фризийски" 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) – ирландски" lang="ga" hreflang="ga" data-title="Feidhm (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="ирландски" 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 – Gan" lang="gan" hreflang="gan" data-title="函數" data-language-autonym="贛語" data-language-local-name="Gan" 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 – галисийски" lang="gl" hreflang="gl" data-title="Función" data-language-autonym="Galego" data-language-local-name="галисийски" 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 – иврит" lang="he" hreflang="he" data-title="פונקציה" data-language-autonym="עברית" data-language-local-name="иврит" 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 – хинди" lang="hi" hreflang="hi" data-title="फलन" data-language-autonym="हिन्दी" data-language-local-name="хинди" 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 – Fiji Hindi" lang="hif" hreflang="hif" data-title="Function" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" 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) – хърватски" lang="hr" hreflang="hr" data-title="Funkcija (matematika)" data-language-autonym="Hrvatski" data-language-local-name="хърватски" 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) – унгарски" lang="hu" hreflang="hu" data-title="Függvény (matematika)" data-language-autonym="Magyar" data-language-local-name="унгарски" 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) – арменски" lang="hy" hreflang="hy" data-title="Ֆունկցիա (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="арменски" 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) – интерлингва" lang="ia" hreflang="ia" data-title="Function (mathematica)" data-language-autonym="Interlingua" data-language-local-name="интерлингва" 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) – индонезийски" lang="id" hreflang="id" data-title="Fungsi (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="индонезийски" 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 – идо" lang="io" hreflang="io" data-title="Funciono" data-language-autonym="Ido" data-language-local-name="идо" 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) – исландски" lang="is" hreflang="is" data-title="Fall (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="исландски" 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) – италиански" lang="it" hreflang="it" data-title="Funzione (matematica)" data-language-autonym="Italiano" data-language-local-name="италиански" 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) – японски" lang="ja" hreflang="ja" data-title="関数 (数学)" data-language-autonym="日本語" data-language-local-name="японски" 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 – ложбан" lang="jbo" hreflang="jbo" data-title="fancu" data-language-autonym="La .lojban." data-language-local-name="ложбан" 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) – грузински" lang="ka" hreflang="ka" data-title="ფუნქცია (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="грузински" 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) – кабилски" lang="kab" hreflang="kab" data-title="Tasɣent (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="кабилски" 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) – казахски" lang="kk" hreflang="kk" data-title="Функция (математика)" data-language-autonym="Қазақша" data-language-local-name="казахски" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%A8%EC%88%98" title="함수 – корейски" lang="ko" hreflang="ko" data-title="함수" data-language-autonym="한국어" data-language-local-name="корейски" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Functio" title="Functio – латински" lang="la" hreflang="la" data-title="Functio" data-language-autonym="Latina" data-language-local-name="латински" 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) – люксембургски" lang="lb" hreflang="lb" data-title="Funktioun (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="люксембургски" 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) – ломбардски" lang="lmo" hreflang="lmo" data-title="Fonzion (matematega)" data-language-autonym="Lombard" data-language-local-name="ломбардски" 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) – лаоски" lang="lo" hreflang="lo" data-title="ຕຳລາ (ຄະນິດສາດ)" data-language-autonym="ລາວ" data-language-local-name="лаоски" 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) – литовски" lang="lt" hreflang="lt" data-title="Funkcija (matematika)" data-language-autonym="Lietuvių" data-language-local-name="литовски" 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 – латвийски" lang="lv" hreflang="lv" data-title="Funkcija" data-language-autonym="Latviešu" data-language-local-name="латвийски" 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) – македонски" lang="mk" hreflang="mk" data-title="Функција (математика)" data-language-autonym="Македонски" data-language-local-name="македонски" 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 – малаялам" lang="ml" hreflang="ml" data-title="ഫങ്ഷൻ" data-language-autonym="മലയാളം" data-language-local-name="малаялам" 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) – монголски" lang="mn" hreflang="mn" data-title="Функц (математик)" data-language-autonym="Монгол" data-language-local-name="монголски" 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) – марати" lang="mr" hreflang="mr" data-title="फल (गणित)" data-language-autonym="मराठी" data-language-local-name="марати" 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 – малайски" lang="ms" hreflang="ms" data-title="Fungsi" data-language-autonym="Bahasa Melayu" data-language-local-name="малайски" 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) – малтийски" lang="mt" hreflang="mt" data-title="Funzjonijiet (matematika)" data-language-autonym="Malti" data-language-local-name="малтийски" 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 – бирмански" lang="my" hreflang="my" data-title="ဖန်ရှင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="бирмански" 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) – долнонемски" lang="nds" hreflang="nds" data-title="Afbillen (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="долнонемски" 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) – нидерландски" lang="nl" hreflang="nl" data-title="Functie (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="нидерландски" 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 – норвежки (нюношк)" lang="nn" hreflang="nn" data-title="Matematisk funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="норвежки (нюношк)" 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) – норвежки (букмол)" lang="nb" hreflang="nb" data-title="Funksjon (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="норвежки (букмол)" 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) – окситански" lang="oc" hreflang="oc" data-title="Aplicacion (matematicas)" data-language-autonym="Occitan" data-language-local-name="окситански" 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) – оромо" lang="om" hreflang="om" data-title="Warroomii (faankishinii)" data-language-autonym="Oromoo" data-language-local-name="оромо" 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) – пенджабски" lang="pa" hreflang="pa" data-title="ਫੰਕਸ਼ਨ (ਹਿਸਾਬ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="пенджабски" 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 – полски" lang="pl" hreflang="pl" data-title="Funkcja" data-language-autonym="Polski" data-language-local-name="полски" 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) – португалски" lang="pt" hreflang="pt" data-title="Função (matemática)" data-language-autonym="Português" data-language-local-name="португалски" 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 – кечуа" lang="qu" hreflang="qu" data-title="Kinraysuyu" data-language-autonym="Runa Simi" data-language-local-name="кечуа" 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 – румънски" lang="ro" hreflang="ro" data-title="Funcție" data-language-autonym="Română" data-language-local-name="румънски" 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) – руски" lang="ru" hreflang="ru" data-title="Функция (математика)" data-language-autonym="Русский" data-language-local-name="руски" 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 – саха" lang="sah" hreflang="sah" data-title="Функция. Функция чэрчитэ, суолталарын түмсээнэ" data-language-autonym="Саха тыла" data-language-local-name="саха" 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) – сицилиански" lang="scn" hreflang="scn" data-title="Funzioni (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="сицилиански" 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) – шотландски" lang="sco" hreflang="sco" data-title="Function (mathematics)" data-language-autonym="Scots" data-language-local-name="шотландски" 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 – сърбохърватски" lang="sh" hreflang="sh" data-title="Funkcija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="сърбохърватски" 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) – словашки" lang="sk" hreflang="sk" data-title="Zobrazenie (matematika)" data-language-autonym="Slovenčina" data-language-local-name="словашки" 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) – словенски" lang="sl" hreflang="sl" data-title="Funkcija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="словенски" 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 – инари-саамски" lang="smn" hreflang="smn" data-title="Funktio" data-language-autonym="Anarâškielâ" data-language-local-name="инари-саамски" 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) – шона" lang="sn" hreflang="sn" data-title="Murimo (Masvomhu)" data-language-autonym="ChiShona" data-language-local-name="шона" 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) – сомалийски" lang="so" hreflang="so" data-title="Shaqada (xisaabta)" data-language-autonym="Soomaaliga" data-language-local-name="сомалийски" 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 – албански" lang="sq" hreflang="sq" data-title="Funksioni" data-language-autonym="Shqip" data-language-local-name="албански" 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) – сръбски" lang="sr" hreflang="sr" data-title="Функција (математика)" data-language-autonym="Српски / srpski" data-language-local-name="сръбски" 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) – сундански" lang="su" hreflang="su" data-title="Fungsi (matematika)" data-language-autonym="Sunda" data-language-local-name="сундански" 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 – шведски" lang="sv" hreflang="sv" data-title="Funktion" data-language-autonym="Svenska" data-language-local-name="шведски" 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 – тамилски" lang="ta" hreflang="ta" data-title="சார்பு" data-language-autonym="தமிழ்" data-language-local-name="тамилски" 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) – тайски" lang="th" hreflang="th" data-title="ฟังก์ชัน (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="тайски" 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) – тагалог" lang="tl" hreflang="tl" data-title="Punsiyon (matematika)" data-language-autonym="Tagalog" data-language-local-name="тагалог" 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 – турски" lang="tr" hreflang="tr" data-title="Fonksiyon" data-language-autonym="Türkçe" data-language-local-name="турски" 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) – татарски" lang="tt" hreflang="tt" data-title="Функция (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="татарски" 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) – удмуртски" lang="udm" hreflang="udm" data-title="Функция (математика)" data-language-autonym="Удмурт" data-language-local-name="удмуртски" 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 – уйгурски" lang="ug" hreflang="ug" data-title="فۇنكسىيە" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="уйгурски" 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) – украински" lang="uk" hreflang="uk" data-title="Функція (математика)" data-language-autonym="Українська" data-language-local-name="украински" 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) – урду" lang="ur" hreflang="ur" data-title="تفاعل (ریاضیات)" data-language-autonym="اردو" data-language-local-name="урду" 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) – узбекски" lang="uz" hreflang="uz" data-title="Funksiya (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="узбекски" 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 – виетнамски" lang="vi" hreflang="vi" data-title="Hàm số" data-language-autonym="Tiếng Việt" data-language-local-name="виетнамски" 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) – варай" lang="war" hreflang="war" data-title="Funsiyon (matematika)" data-language-autonym="Winaray" data-language-local-name="варай" 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 – ву китайски" lang="wuu" hreflang="wuu" data-title="函数" data-language-autonym="吴语" data-language-local-name="ву китайски" 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 – калмик" lang="xal" hreflang="xal" data-title="Функция" data-language-autonym="Хальмг" data-language-local-name="калмик" 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 – идиш" lang="yi" hreflang="yi" data-title="פונקציע" data-language-autonym="ייִדיש" data-language-local-name="идиш" 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) – стандартен марокански тамазигт" lang="zgh" hreflang="zgh" data-title="ⵜⴰⵙⵖⵏⵜ (ⵜⵓⵙⵏⴰⴽⵜ)" data-language-autonym="ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ" data-language-local-name="стандартен марокански тамазигт" 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 – китайски" lang="zh" hreflang="zh" data-title="函数" data-language-autonym="中文" data-language-local-name="китайски" 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 – Minnan" lang="nan" hreflang="nan" data-title="Hâm-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" 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 – кантонски" lang="yue" hreflang="yue" data-title="函數" data-language-autonym="粵語" data-language-local-name="кантонски" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11348#sitelinks-wikipedia" title="Редактиране на междуезиковите препратки" class="wbc-editpage">Редактиране</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Именни пространства"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Преглед на основната страница [c]" accesskey="c"><span>Статия</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%D0%91%D0%B5%D1%81%D0%B5%D0%B4%D0%B0:%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" rel="discussion" title="Беседа за страницата [t]" accesskey="t"><span>Беседа</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Промяна на езиковия вариант" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">български</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div 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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">от Уикипедия, свободната енциклопедия</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="bg" dir="ltr"><div class="othermeaning-box"> <dl><dd><i>Вижте <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_(%D0%BF%D0%BE%D1%8F%D1%81%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5)" class="mw-disambig" title="Функция (пояснение)">пояснителната страница</a> за други значения на <b>Функция</b>.</i></dd></dl> <hr /></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Graph_of_example_function.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Graph_of_example_function.svg/250px-Graph_of_example_function.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Graph_of_example_function.svg/375px-Graph_of_example_function.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Graph_of_example_function.svg/500px-Graph_of_example_function.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Графика на функцията<br /><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 {\begin{aligned}&\scriptstyle \\&\textstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mstyle displaystyle="false" scriptlevel="1" /> </mtd> </mtr> <mtr> <mtd /> <mtd> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mrow> <mn>3</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\scriptstyle \\&\textstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849e95193ac47959d04871bd7819452dc57455b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.945ex; margin-bottom: -0.227ex; width:23.359ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}&\scriptstyle \\&\textstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}}"></span></figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Graph_of_function_of_2_variables.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Graph_of_function_of_2_variables.png/250px-Graph_of_function_of_2_variables.png" decoding="async" width="250" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Graph_of_function_of_2_variables.png/375px-Graph_of_function_of_2_variables.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Graph_of_function_of_2_variables.png/500px-Graph_of_function_of_2_variables.png 2x" data-file-width="551" data-file-height="447" /></a><figcaption>Графика на функцията на две променливи<br /><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)=\sin(\pi \times {\sqrt {x^{2}+y^{2}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=\sin(\pi \times {\sqrt {x^{2}+y^{2}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c1731fcfd35296d5cce58b3c6e04826e27e88f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:28.306ex; height:4.843ex;" alt="{\displaystyle f(x,y)=\sin(\pi \times {\sqrt {x^{2}+y^{2}}})}"></span></figcaption></figure> <p><b>Функция</b> в <a href="/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика">математиката</a> е съпоставяне на определена величина, наричана <i>аргумент</i>, на друга величина, наричана <i>стойност</i>, като на всеки аргумент се съпоставя точно една стойност. Аргументът и стойността могат да бъдат <a href="/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Реално число">реални числа</a>, но също и елементи на всяко друго <a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество">множество</a>. Пример за функция е <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)=2x}"> <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> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a8ebea86ba5d3a71121e0a4156f5ec07b25220" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.008ex; height:2.843ex;" alt="{\displaystyle f(x)=2x}"></span> – функция, която съпоставя на всяко число числото, два пъти по-голямо от него. Така на 5 се съпоставя 10, което се изписва като <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(5)=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(5)=10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b4f69a76ea2702bc7decf7b679dd1930bb0035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.674ex; height:2.843ex;" alt="{\displaystyle f(5)=10}"></span>. </p><p>Аргументите<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> на функциите могат да бъдат не само числа, но и други добре определени обекти. Например дадена функция може да съпоставя на буквата A числото 1, на буквата B числото 2 и така нататък. Съществуват много начини за описване или представяне на функциите – <a href="/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0" title="Формула">формули</a>, <a href="/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D1%8A%D0%BC" title="Алгоритъм">алгоритми</a>, изчисляващи стойностите за различни аргументи, <a href="/wiki/%D0%93%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Графика на функция">графики</a>, които дават графично изображение на стойностите, или таблици със стойностите за конкретни аргументи, често използвани в <a href="/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0" title="Статистика">статистиката</a>, природните науки и <a href="/wiki/%D0%A2%D0%B5%D1%85%D0%BD%D0%B8%D0%BA%D0%B0" title="Техника">техниката</a>. </p><p>Множеството от всички възможни стойности на аргументите на дадена функция се нарича дефиниционно множество, дефиниционна област или <a href="/wiki/%D0%9E%D0%B1%D0%BB%D0%B0%D1%81%D1%82_%D0%BD%D0%B0_%D0%BE%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Област на определение на функция">област на определение на функция</a>. В съвременната математика функциите обикновено се дефинират и с определено множество, включващо всички възможни стойности на функцията. Например функциите с реални стойности имат за такова множество всички реални числа, дори когато отделни такива функции не включват всяко реално число сред своите стойности. </p><p>Функциите могат да бъдат описани и чрез отношението си към други функции. Например като <a href="/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Обратна функция">обратната функция</a> на дадена функция или като решението на <a href="/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%BE_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Диференциално уравнение">диференциално уравнение</a>. Функциите могат за бъдат събирани, умножавани или съчетавани по други начини, за да се получат нови функции. Важно действие, извършвано върху функциите, което ги отличава от <a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE" title="Число">числата</a>, е <a href="/w/index.php?title=%D0%9A%D0%BE%D0%BC%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D1%8F_%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8&action=edit&redlink=1" class="new" title="Композиция на функции (страницата не съществува)">композицията</a>, при която стойността на дадена функция става аргумент на друга функция. Групи функции с определени свойства, например <a href="/w/index.php?title=%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D0%BA%D1%8A%D1%81%D0%BD%D0%B0%D1%82%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Непрекъсната функция (страницата не съществува)">непрекъснати функции</a> или <a href="/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%80%D1%83%D0%B5%D0%BC%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" class="mw-redirect" title="Диференцируема функция">диференцируеми функции</a>, се наричат <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Функционно пространство">функционни пространства</a> и се изследват като самостоятелни обекти в области като <a href="/wiki/%D0%A0%D0%B5%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Реален анализ">реалния</a> и <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Комплексен анализ">комплексния анализ</a>. </p><p>Съществуват <a href="/w/index.php?title=%D0%9D%D0%B5%D0%B8%D0%B7%D0%B1%D1%80%D0%BE%D0%B8%D0%BC%D0%BE_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE&action=edit&redlink=1" class="new" title="Неизброимо множество (страницата не съществува)">неизброимо много</a> различни функции, повечето от които не могат да бъдат описани с формула или алгоритъм. Строга дефиниция на понятието <i>функция</i> може да бъде формулирано в <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D1%82%D0%B0" title="Теория на множествата">теорията на множествата</a> с помощта на <a href="/wiki/%D0%9D%D0%B0%D1%80%D0%B5%D0%B4%D0%B5%D0%BD%D0%B0_%D0%B4%D0%B2%D0%BE%D0%B9%D0%BA%D0%B0" title="Наредена двойка">наредени двойки</a> и <a href="/wiki/%D0%A0%D0%B5%D0%BB%D0%B0%D1%86%D0%B8%D1%8F" title="Релация">релации</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Терминология"><span id=".D0.A2.D0.B5.D1.80.D0.BC.D0.B8.D0.BD.D0.BE.D0.BB.D0.BE.D0.B3.D0.B8.D1.8F"></span>Терминология</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=1" title="Редактиране на раздел: Терминология" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=1" title="Edit section's source code: Терминология"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Codomain2.SVG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/250px-Codomain2.SVG.png" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/375px-Codomain2.SVG.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Codomain2.SVG/500px-Codomain2.SVG.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption>Дефиниционното множество X, множеството на стойностите f(x) и включващото го множество Y</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Function_machine2-bg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Function_machine2-bg.svg/250px-Function_machine2-bg.svg.png" decoding="async" width="250" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Function_machine2-bg.svg/375px-Function_machine2-bg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Function_machine2-bg.svg/500px-Function_machine2-bg.svg.png 2x" data-file-width="159" data-file-height="152" /></a><figcaption>Функцията ƒ има аргумент <i>x</i> и стойност ƒ(<i>x</i>) – метафорично тя може да бъде описана като устройство, което преобразува аргумента в стойност</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Fun%C3%A7%C3%A3o_cotg(x).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fun%C3%A7%C3%A3o_cotg%28x%29.png/250px-Fun%C3%A7%C3%A3o_cotg%28x%29.png" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fun%C3%A7%C3%A3o_cotg%28x%29.png/375px-Fun%C3%A7%C3%A3o_cotg%28x%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fun%C3%A7%C3%A3o_cotg%28x%29.png/500px-Fun%C3%A7%C3%A3o_cotg%28x%29.png 2x" data-file-width="1200" data-file-height="900" /></a><figcaption>Графика на прекъсната функция <i>cotg</i>(<i>x</i>)</figcaption></figure> <p>Функциите се срещат във всички области на математиката и природните науки, но различните области имат различни означения, различна представа за свойствата на функциите и дори различна дефиниция. <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%B0%D1%82%D0%B0" title="Теория на множествата">Теорията на множествата</a> разглежда функциите в най-голяма общност. Единственото свойство, което се изисква от една функция, е да съпоставя единствена стойност на всеки свой допустим аргумент. Не се изисква аргументът или стойността да са <a href="/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE" title="Число">числа</a>, например функцията, която съпоставя на всяка държава нейната столица, не задава зависимост между числови <a href="/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество">множества</a>. В <a href="/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Алгебра">алгебрата</a> функциите обикновено се изразяват с помощта на алгебрични операции. </p><p>Функциите, изследвани в <a href="/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Математически анализ">анализа</a>, обикновено притежават допълнителни свойства като <a href="/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D0%BA%D1%8A%D1%81%D0%BD%D0%B0%D1%82%D0%BE%D1%81%D1%82" title="Непрекъснатост">непрекъснатост</a> или <a href="/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%80%D1%83%D0%B5%D0%BC%D0%BE%D1%81%D1%82" class="mw-redirect" title="Диференцируемост">диференцируемост</a>. Пример за такава функция е функцията <a href="/wiki/%D0%A1%D0%B8%D0%BD%D1%83%D1%81_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" class="mw-redirect" title="Синус (математика)">синус</a>. Обикновено изучаваните там функции не могат да се изразят с една-единствена формула. В <a href="/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%BB%D0%B5%D0%BA%D1%81%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Комплексен анализ">комплексния анализ</a> се разглеждат <a href="/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Аналитична функция">аналитични функции</a>, които могат да се изразят чрез развитие в <a href="/wiki/%D0%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D0%B5%D0%BD_%D1%80%D0%B5%D0%B4" title="Степенен ред">степенен ред</a>. В комплексния анализ се разглеждат и специален клас <a href="/w/index.php?title=%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%B7%D0%BD%D0%B0%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Многозначна функция (страницата не съществува)">многозначни функции</a>, които могат да съпоставят повече от една стойност на даден аргумент. Въпреки че формално погледнато те не са функции, те имат много близки свойства до свойствата на аналитичните функции. За разлика от теорията на множествата в <a href="/w/index.php?title=%D0%9B%D0%B0%D0%BC%D0%B1%D0%B4%D0%B0-%D1%81%D0%BC%D1%8F%D1%82%D0%B0%D0%BD%D0%B5&action=edit&redlink=1" class="new" title="Ламбда-смятане (страницата не съществува)">ламбда-смятането</a> функциите са примитивен обект и не се дефинират посредством множества. </p><p>В много области на математиката термините <i>карта</i>, <i>изображение</i>, <i>трансформация</i> и <i>оператор</i> се използват като синоними на <i>функция</i>. В някои случаи обаче те могат да имат по-специално значение. Например под <i>трансформация</i> често се разбира функция, за която множеството на аргументите и множеството на стойностите съвпадат. В <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8%D1%82%D0%B5" title="Теория на категориите">теорията на категориите</a> се използва понятието <i>морфизъм</i>, което е обобщение на някои видове функции. </p><p>Аргументът на функцията, наричан също <i>независима променлива</i>, най-често се означава с буквата <i>x</i> или, когато изразява <a href="/wiki/%D0%92%D1%80%D0%B5%D0%BC%D0%B5" title="Време">време</a>, с буквата <i>t</i>. Стойността на функцията обикновено се изразява с буквата <i>y</i>. За самата функция в повечето случаи се използва символът <i>f</i>. Така изразът <i>y</i> = <i>f</i>(<i>x</i>) показва, че функцията, наречена <i>f</i>, има аргумент, наречен <i>x</i>, и стойност, наречена <i>y</i>. </p><p>Множеството от всички позволени аргументи на дадена функция се нарича нейно <a href="/wiki/%D0%94%D0%B5%D1%84%D0%B8%D0%BD%D0%B8%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%BE_%D0%BC%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" class="mw-redirect" title="Дефиниционно множество">дефиниционно множество</a>, а множеството от всички стойности – <a href="/w/index.php?title=%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE_%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D0%B9%D0%BD%D0%BE%D1%81%D1%82%D0%B8%D1%82%D0%B5&action=edit&redlink=1" class="new" title="Множество на стойностите (страницата не съществува)">множество на стойностите</a> на функцията. Така функцията <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> има за дефиниционно множество всички реални числа, а множеството на стойностите ̀ включва всички неотрицателни реални числа. </p> <div class="mw-heading mw-heading2"><h2 id="Формални_дефиниции"><span id=".D0.A4.D0.BE.D1.80.D0.BC.D0.B0.D0.BB.D0.BD.D0.B8_.D0.B4.D0.B5.D1.84.D0.B8.D0.BD.D0.B8.D1.86.D0.B8.D0.B8"></span>Формални дефиниции</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=2" title="Редактиране на раздел: Формални дефиниции" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=2" title="Edit section's source code: Формални дефиниции"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Една приблизителна дефиниция на понятието <i>функция</i> е следната: Нека <i>A</i> и <i>B</i> са множества. Функция от <i>A</i> в <i>B</i> е правило, което съпоставя на всеки елемент от A точно един елемент от <i>B</i>. Тази интуитивна представа за функциите се използва от древни времена и все още се среща на места, където строга дефиниция не е необходима, например в училищните учебници по математика. Проблемът при нея е, че зависи от неясното понятие <i>правило</i>. </p><p>Поради това в теорията на множествата се използва следната дефиниция: <b>Частична функция</b> <i>f</i> от множество <i>A</i> в множество <i>B</i> се нарича всяко подмножество <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_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86fb9e8d359d1eacb9be564288c4f2f64ce44ab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.963ex; height:2.843ex;" alt="{\displaystyle G_{f}}"></span> на <a href="/wiki/%D0%94%D0%B5%D0%BA%D0%B0%D1%80%D1%82%D0%BE%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Декартово произведение">декартовото произведение</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\times B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f31ae45b0098f06b5d22c38d317eb097a88fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.348ex; height:2.176ex;" alt="{\displaystyle A\times B}"></span> такова, че за всяко <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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bcc9b2afb295d4234bc294860cd0c63bcad2ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle x\in A}"></span> съществува най-много едно <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ccabd006952897bb52668533010cb9e4ab3f77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.76ex; height:2.509ex;" alt="{\displaystyle y\in B}"></span> за което <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 G_{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> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\in G_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29cd3829ab31f48347510ce1511f93892455ef34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.132ex; height:3.009ex;" alt="{\displaystyle (x,y)\in G_{f}}"></span>. Ако за всяко <i>x</i> съществува точно едно такова <i>y</i>, то <i>f</i> се нарича <b>тотална функция</b>. Под <b>функция</b> обикновено се разбира <i>тотална функция</i>. Множеството <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_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86fb9e8d359d1eacb9be564288c4f2f64ce44ab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.963ex; height:2.843ex;" alt="{\displaystyle G_{f}}"></span> се нарича <i>графика на функцията</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Функция-оригинал"><span id=".D0.A4.D1.83.D0.BD.D0.BA.D1.86.D0.B8.D1.8F-.D0.BE.D1.80.D0.B8.D0.B3.D0.B8.D0.BD.D0.B0.D0.BB"></span>Функция-оригинал</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=3" title="Редактиране на раздел: Функция-оригинал" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=3" title="Edit section's source code: Функция-оригинал"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Функция-оригинал</b> или <b>оригинална функция</b> е фундаментално понятие в операционното изчисление; за да се нарече функцията <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 {C} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <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">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon {\mathbb {R} \to \mathbb {C} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afc7e8d6eca1b0a97e4b52389a8ab7970278c63" 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 {C} }}"></span> оригинал, тя трябва да отговаря на три условия: </p> <ol><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> да бъде <a href="/wiki/%D0%9B%D0%B8%D0%BF%D1%88%D0%B8%D1%86%D0%BE%D0%B2%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Липшицова функция">Липшицова функция</a> почти навсякъде по реалната права <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>, освен това във всички точки в произволен краен интервал <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;b)\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>;</mo> <mi>b</mi> <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 (a;b)\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b357fde5506073d0e1dbef35d455e9d204cced3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.847ex; height:2.843ex;" alt="{\displaystyle (a;b)\subset \mathbb {R} }"></span>, в които определеното условие не е изпълнено, функцията трябва да претърпи <a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D1%80%D1%8B%D0%B2%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F#Точки_разрыва" class="extiw" title="ru:Непрерывная функция">прекъсване от 1-ви род</a>. Формално, за произволно <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>, което не принадлежи на споменатото множество, трябва да съществуват положителни константи <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,\,\alpha \leqslant 1,\,h_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>α</mi> <mo>⩽</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,\,\alpha \leqslant 1,\,h_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a2b61d7189324ec5efb07f73341f85ac13988d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.727ex; height:2.509ex;" alt="{\displaystyle A,\,\alpha \leqslant 1,\,h_{0}}"></span>, така че <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(t+h)-f(t)|\leqslant A|h|^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩽</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(t+h)-f(t)|\leqslant A|h|^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11cfe6b308731503127e84b0bda372fff36e6230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.927ex; height:3.009ex;" alt="{\displaystyle |f(t+h)-f(t)|\leqslant A|h|^{\alpha }}"></span> за произволно <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 h\in [-h_{0};h_{0}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>;</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in [-h_{0};h_{0}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac70ccdff9ab050d6436b09b9669cad3e675c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.102ex; height:2.843ex;" alt="{\displaystyle h\in [-h_{0};h_{0}]}"></span>.</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(t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1b63ab4c491516b6e1444f97dac7ab5e439d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.188ex; height:2.843ex;" alt="{\displaystyle f(t)=0}"></span> за <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 t<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c8875f14d87cb6daa44307512a91eceb5f34d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t<0}"></span>.</li> <li>На функцията <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(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> се налага определено ограничение - тя не трябва да нараства по-бързо от <a href="/wiki/%D0%95%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Експоненциална функция">експоненциалната функция</a>. Формално, за тази функция трябва да съществуват константи <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 M>0,\,s_{0}\geqslant 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⩾</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M>0,\,s_{0}\geqslant 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d403506363180573be37954fb7e0ac25908ead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.53ex; height:2.509ex;" alt="{\displaystyle M>0,\,s_{0}\geqslant 0}"></span>, така че <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(t)|<Me^{s_{0}t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>M</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |f(t)|<Me^{s_{0}t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9029532288281a3f14fc4a0c0e7576537c985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.274ex; height:3.009ex;" alt="{\displaystyle |f(t)|<Me^{s_{0}t}}"></span> за произволно <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 t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592bced0c39b10fc90e74c6a66223abfbfb029de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {R} }"></span>.</li></ol> <p>За повечето физически проблеми и трите условия са изпълнени. Освен това, използвайки <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%A5%D0%B5%D0%B2%D0%B8%D1%81%D0%B0%D0%B9%D0%B4" title="Функция на Хевисайд">функцията на Хевисайд</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 H(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1b6c8837aed2794e7b52afa88ad371f1d275fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:2.843ex;" alt="{\displaystyle H(t)}"></span> е възможно да се получи оригиналната функция от функция, която отговаря само на условия 1 и 3. </p> <div class="mw-heading mw-heading2"><h2 id="История_на_понятието"><span id=".D0.98.D1.81.D1.82.D0.BE.D1.80.D0.B8.D1.8F_.D0.BD.D0.B0_.D0.BF.D0.BE.D0.BD.D1.8F.D1.82.D0.B8.D0.B5.D1.82.D0.BE"></span>История на понятието</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=4" title="Редактиране на раздел: История на понятието" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=4" title="Edit section's source code: История на понятието"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Обекти, които според съвременните разбирания се считат за функции, са били разглеждани още в дълбока древност. В <a href="/wiki/%D0%92%D0%B0%D0%B2%D0%B8%D0%BB%D0%BE%D0%BD%D0%B8%D1%8F" title="Вавилония">Древен Вавилон</a> например са открити таблици на квадратите и кубовете на естествените числа. <a href="/wiki/%D0%9F%D1%82%D0%BE%D0%BB%D0%B5%D0%BC%D0%B5%D0%B9" class="mw-redirect" title="Птолемей">Птолемей</a> е изчислявал дължини на хорди в окръжност, което по същество означава, че е използвал <a href="/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Тригонометрична функция">тригонометрични функции</a>. Понятието обаче започва да се оформя през XIV век. Самото название <i>функция</i> се използва за първи път от <a href="/wiki/%D0%93%D0%BE%D1%82%D1%84%D1%80%D0%B8%D0%B4_%D0%9B%D0%B0%D0%B9%D0%B1%D0%BD%D0%B8%D1%86" title="Готфрид Лайбниц">Готфрид Лайбниц</a> около <a href="/wiki/1670" title="1670">1670</a> г. Функциите, които той е разглеждал, днес се наричат <a href="/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%80%D1%83%D0%B5%D0%BC%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" class="mw-redirect" title="Диференцируема функция">диференцируеми функции</a> и са най-често срещаният вид функции в приложенията на математиката. За тях имат смисъл понятията <a href="/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Граница (математика)">граница</a> и <a href="/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D0%B0" title="Производна">производна</a>. </p><p>През <a href="/wiki/1755" title="1755">1755</a> г. <a href="/wiki/%D0%9B%D0%B5%D0%BE%D0%BD%D0%B0%D1%80%D0%B4_%D0%9E%D0%B9%D0%BB%D0%B5%D1%80" title="Леонард Ойлер">Леонард Ойлер</a> дава в книгата си <i>Institutiones calculi differentialis</i> съвременното разбиране за функция, а именно зависимост между две величини, при което промяната на едната величина (аргумента на функцията) води до промяна на другата величина (стойността на функцията). Въпреки това определение обаче Ойлер разглежда само <a href="/w/index.php?title=%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D0%BA%D1%8A%D1%81%D0%BD%D0%B0%D1%82%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Непрекъсната функция (страницата не съществува)">непрекъснати функции</a>, които могат да се изразят с формула, състояща се от крайно или безкрайно много алгебрични операции. <a href="/wiki/%D0%96%D0%B0%D0%BD_%D0%91%D0%B0%D1%82%D0%B8%D1%81%D1%82_%D0%96%D0%BE%D0%B7%D0%B5%D1%84_%D0%A4%D1%83%D1%80%D0%B8%D0%B5" class="mw-redirect" title="Жан Батист Жозеф Фурие">Фурие</a> започва да раглежда и някои прекъснати функции, но той смята, че всяка функция може да се изрази чрез <a href="/wiki/%D0%A0%D0%B5%D0%B4_%D0%BD%D0%B0_%D0%A4%D1%83%D1%80%D0%B8%D0%B5" title="Ред на Фурие">ред на Фурие</a>. <a href="/wiki/%D0%9F%D0%B5%D1%82%D0%B5%D1%80_%D0%93%D1%83%D1%81%D1%82%D0%B0%D0%B2_%D0%9B%D1%8C%D0%BE%D0%B6%D0%BE%D0%BD_%D0%94%D0%B8%D1%80%D0%B8%D1%85%D0%BB%D0%B5" title="Петер Густав Льожон Дирихле">Дирихле</a> за пръв път разглежда числовите функции в пълната им общност. Той дава съвременната дефиниция на непрекъсната функция и дава пример за навсякъде прекъсната функция. Също така изяснява разликата между функцията и нейното представяне чрез формули. </p> <div class="mw-heading mw-heading2"><h2 id="Видове_функции"><span id=".D0.92.D0.B8.D0.B4.D0.BE.D0.B2.D0.B5_.D1.84.D1.83.D0.BD.D0.BA.D1.86.D0.B8.D0.B8"></span>Видове функции</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=5" title="Редактиране на раздел: Видове функции" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=5" title="Edit section's source code: Видове функции"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/17px-Vista-xmag.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/26px-Vista-xmag.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/34px-Vista-xmag.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Вижте също: <a href="/wiki/%D0%92%D0%B4%D0%BB%D1%8A%D0%B1%D0%BD%D0%B0%D1%82%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Вдлъбната функция">Вдлъбната функция</a>, <a href="/wiki/%D0%98%D0%B7%D0%BF%D1%8A%D0%BA%D0%BD%D0%B0%D0%BB%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Изпъкнала функция">Изпъкнала функция</a> и <a href="/wiki/%D0%9E%D0%B1%D1%80%D0%B0%D1%82%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Обратна функция">Обратна функция</a></i></div></dd> <dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/17px-Vista-xmag.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/26px-Vista-xmag.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/34px-Vista-xmag.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Вижте също: <a href="/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B8%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Хиперболична функция">Хиперболична функция</a>, <a href="/wiki/%D0%95%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Експоненциална функция">Експоненциална функция</a> и <a href="/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D0%BE%D0%BD%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Тригонометрична функция">Тригонометрична функция</a></i></div></dd> <dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/17px-Vista-xmag.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/26px-Vista-xmag.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/34px-Vista-xmag.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Вижте също: <a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Специални функции">Специални функции</a>, <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%92%D0%B0%D0%B9%D0%B5%D1%80%D1%89%D1%80%D0%B0%D1%81" title="Функция на Вайерщрас">Функция на Вайерщрас</a>, <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%91%D0%B5%D1%81%D0%B5%D0%BB" title="Функция на Бесел">Функция на Бесел</a> и <a href="/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%A5%D0%B5%D0%B2%D0%B8%D1%81%D0%B0%D0%B9%D0%B4" title="Функция на Хевисайд">Функция на Хевисайд</a></i></div></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Свойства_на_функциите"><span id=".D0.A1.D0.B2.D0.BE.D0.B9.D1.81.D1.82.D0.B2.D0.B0_.D0.BD.D0.B0_.D1.84.D1.83.D0.BD.D0.BA.D1.86.D0.B8.D0.B8.D1.82.D0.B5"></span>Свойства на функциите</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=6" title="Редактиране на раздел: Свойства на функциите" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=6" title="Edit section's source code: Свойства на функциите"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Дефинируемост"><span id=".D0.94.D0.B5.D1.84.D0.B8.D0.BD.D0.B8.D1.80.D1.83.D0.B5.D0.BC.D0.BE.D1.81.D1.82"></span>Дефинируемост</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=7" title="Редактиране на раздел: Дефинируемост" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=7" title="Edit section's source code: Дефинируемост"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 y=2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/582d8299e827a8ee042bff79fd37ead41199f7ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle y=2x}"></span> е дефинирана за всяко <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> в <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 ,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.299ex; height:2.843ex;" alt="{\displaystyle (-\infty ,\infty )}"></span>, но <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={\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0871f6114093b98d171970f2974952524b02d1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.42ex; height:5.176ex;" alt="{\displaystyle y={\frac {1}{x}}}"></span> е дефинирана за <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 ,0)\cup (0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∪</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 (-\infty ,0)\cup (0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c52e6986a31c45dfa48d61b80960c6be429725e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.05ex; height:2.843ex;" alt="{\displaystyle (-\infty ,0)\cup (0,\infty )}"></span>, тъй като трябва <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\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Диференцируемост"><span id=".D0.94.D0.B8.D1.84.D0.B5.D1.80.D0.B5.D0.BD.D1.86.D0.B8.D1.80.D1.83.D0.B5.D0.BC.D0.BE.D1.81.D1.82"></span>Диференцируемост</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=8" title="Редактиране на раздел: Диференцируемост" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=8" title="Edit section's source code: Диференцируемост"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/17px-Vista-xmag.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/26px-Vista-xmag.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Vista-xmag.png/34px-Vista-xmag.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Вижте също: <a href="/wiki/%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Граница (математика)">Граница (математика)</a> и <a href="/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D0%B0" title="Производна">Производна</a></i></div></dd></dl> <p>Ако границата <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_{0})=\lim _{x\to x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x_{0})=\lim _{x\to x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc39829b7f50a49f30244f0a7c57aa55e07bac2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.605ex; height:6.009ex;" alt="{\displaystyle f'(x_{0})=\lim _{x\to x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}}"></span> съществува, казваме, че функцията е диференцируема в точка <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>. </p><p>Например да вземем функцията <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=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" 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 y=x}"></span>, тогава <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_{0})=\lim _{x\to x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}={\frac {x-x_{0}}{x-x_{0}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x_{0})=\lim _{x\to x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}={\frac {x-x_{0}}{x-x_{0}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/331bcb6b1073432263f34acfb0ab3adca8855c34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.355ex; height:6.009ex;" alt="{\displaystyle f'(x_{0})=\lim _{x\to x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}={\frac {x-x_{0}}{x-x_{0}}}=1}"></span>. Интуитивно се вижда, че тази функция описва права, която преминава през координатното начало и освен това е ъглополовяща, сиреч сключва ъгъл 45 градуса с абсцисната ос, а оттам <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 \tan \alpha =\tan 45^{\circ }=1=f'(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan \alpha =\tan 45^{\circ }=1=f'(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b2491f8aca40e325465bd1fb796b9aa24ba60d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.962ex; height:3.009ex;" alt="{\displaystyle \tan \alpha =\tan 45^{\circ }=1=f'(x)}"></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Непрекъснатост"><span id=".D0.9D.D0.B5.D0.BF.D1.80.D0.B5.D0.BA.D1.8A.D1.81.D0.BD.D0.B0.D1.82.D0.BE.D1.81.D1.82"></span>Непрекъснатост</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=9" title="Редактиране на раздел: Непрекъснатост" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=9" title="Edit section's source code: Непрекъснатост"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/17px-Crystal_Clear_app_xmag.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/26px-Crystal_Clear_app_xmag.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/34px-Crystal_Clear_app_xmag.svg.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Основна статия: <a href="/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D0%BA%D1%8A%D1%81%D0%BD%D0%B0%D1%82%D0%BE%D1%81%D1%82" title="Непрекъснатост">Непрекъснатост</a></i></div></dd></dl> <p><b>Теорема:</b> Ако функцията <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>е диференцируема в <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>, то тя е непрекъсната в <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>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Интегруемост"><span id=".D0.98.D0.BD.D1.82.D0.B5.D0.B3.D1.80.D1.83.D0.B5.D0.BC.D0.BE.D1.81.D1.82"></span>Интегруемост</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=10" title="Редактиране на раздел: Интегруемост" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=10" title="Edit section's source code: Интегруемост"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Интегруемост_в_интервал"><span id=".D0.98.D0.BD.D1.82.D0.B5.D0.B3.D1.80.D1.83.D0.B5.D0.BC.D0.BE.D1.81.D1.82_.D0.B2_.D0.B8.D0.BD.D1.82.D0.B5.D1.80.D0.B2.D0.B0.D0.BB"></span>Интегруемост в интервал</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=11" title="Редактиране на раздел: Интегруемост в интервал" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=11" title="Edit section's source code: Интегруемост в интервал"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ако интегралът <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 \int _{a}^{b}f\left(x\right)\,dx=\lim _{\lambda \to 0}\sigma =\lim _{\lambda \to 0}\sum _{i=1}^{n}f(\xi _{i})(x_{i}-x_{i-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mi>σ</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</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>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f\left(x\right)\,dx=\lim _{\lambda \to 0}\sigma =\lim _{\lambda \to 0}\sum _{i=1}^{n}f(\xi _{i})(x_{i}-x_{i-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edba3518b1cb40b76a325d8d5a26d7293409b6fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.718ex; height:7.009ex;" alt="{\displaystyle \int _{a}^{b}f\left(x\right)\,dx=\lim _{\lambda \to 0}\sigma =\lim _{\lambda \to 0}\sum _{i=1}^{n}f(\xi _{i})(x_{i}-x_{i-1})}"></span> съществува, то се казва, че <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> е интегруема в <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,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span>, а <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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> и <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/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> се наричат долна и горна граница на интеграла.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Монотонност"><span id=".D0.9C.D0.BE.D0.BD.D0.BE.D1.82.D0.BE.D0.BD.D0.BD.D0.BE.D1.81.D1.82"></span>Монотонност</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=12" title="Редактиране на раздел: Монотонност" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=12" title="Edit section's source code: Монотонност"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/17px-Crystal_Clear_app_xmag.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/26px-Crystal_Clear_app_xmag.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/34px-Crystal_Clear_app_xmag.svg.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Основна статия: <a href="/wiki/%D0%9C%D0%BE%D0%BD%D0%BE%D1%82%D0%BE%D0%BD%D0%BD%D0%BE%D1%81%D1%82" class="mw-redirect" title="Монотонност">Монотонност</a></i></div></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Графика_на_функция"><span id=".D0.93.D1.80.D0.B0.D1.84.D0.B8.D0.BA.D0.B0_.D0.BD.D0.B0_.D1.84.D1.83.D0.BD.D0.BA.D1.86.D0.B8.D1.8F"></span>Графика на функция</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=13" title="Редактиране на раздел: Графика на функция" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=13" title="Edit section's source code: Графика на функция"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/17px-Crystal_Clear_app_xmag.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/26px-Crystal_Clear_app_xmag.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/34px-Crystal_Clear_app_xmag.svg.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Основна статия: <a href="/wiki/%D0%93%D1%80%D0%B0%D1%84%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Графика на функция">Графика на функция</a></i></div></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Екстремуми_на_функция"><span id=".D0.95.D0.BA.D1.81.D1.82.D1.80.D0.B5.D0.BC.D1.83.D0.BC.D0.B8_.D0.BD.D0.B0_.D1.84.D1.83.D0.BD.D0.BA.D1.86.D0.B8.D1.8F"></span>Екстремуми на функция</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=14" title="Редактиране на раздел: Екстремуми на функция" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=14" title="Edit section's source code: Екстремуми на функция"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/17px-Crystal_Clear_app_xmag.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/26px-Crystal_Clear_app_xmag.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/34px-Crystal_Clear_app_xmag.svg.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Основна статия: <a href="/wiki/%D0%9B%D0%BE%D0%BA%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B5%D0%BA%D1%81%D1%82%D1%80%D0%B5%D0%BC%D1%83%D0%BC" class="mw-redirect" title="Локален екстремум">Локален екстремум</a></i></div></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Инфлексна_точка"><span id=".D0.98.D0.BD.D1.84.D0.BB.D0.B5.D0.BA.D1.81.D0.BD.D0.B0_.D1.82.D0.BE.D1.87.D0.BA.D0.B0"></span>Инфлексна точка</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=15" title="Редактиране на раздел: Инфлексна точка" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=15" title="Edit section's source code: Инфлексна точка"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="dablink noprint"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/17px-Crystal_Clear_app_xmag.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/26px-Crystal_Clear_app_xmag.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Crystal_Clear_app_xmag.svg/34px-Crystal_Clear_app_xmag.svg.png 2x" data-file-width="128" data-file-height="128" /></span></span> <i>Основна статия: <a href="/wiki/%D0%98%D0%BD%D1%84%D0%BB%D0%B5%D0%BA%D1%81%D0%BD%D0%B0_%D1%82%D0%BE%D1%87%D0%BA%D0%B0" title="Инфлексна точка">Инфлексна точка</a></i></div></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Вижте_също"><span id=".D0.92.D0.B8.D0.B6.D1.82.D0.B5_.D1.81.D1.8A.D1.89.D0.BE"></span>Вижте също</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=16" title="Редактиране на раздел: Вижте също" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=16" title="Edit section's source code: Вижте също"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D0%98%D0%B7%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" class="mw-redirect" title="Изображение (алгебра)">Изображение (алгебра)</a></li> <li><a href="/w/index.php?title=%D0%93%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0_%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&redlink=1" class="new" title="Граница на функция (страницата не съществува)">Граница на функция</a></li> <li><a href="/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%80%D1%83%D0%B5%D0%BC%D0%BE%D1%81%D1%82" class="mw-redirect" title="Диференцируемост">Диференцируемост</a></li> <li><a href="/wiki/%D0%A2%D0%BE%D1%87%D0%BA%D0%B0_%D0%BD%D0%B0_%D0%BF%D1%80%D0%B5%D0%BA%D1%8A%D1%81%D0%B2%D0%B0%D0%BD%D0%B5" title="Точка на прекъсване">Точка на прекъсване</a></li> <li><a href="/wiki/%D0%9C%D0%BE%D0%BD%D0%BE%D1%82%D0%BE%D0%BD%D0%BD%D0%BE%D1%81%D1%82" class="mw-redirect" title="Монотонност">Монотонност</a></li> <li><a href="/w/index.php?title=%D0%A5%D0%BE%D0%BC%D0%BE%D0%B3%D0%B5%D0%BD%D0%BD%D0%BE%D1%81%D1%82_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)&action=edit&redlink=1" class="new" title="Хомогенност (математика) (страницата не съществува)">Хомогенност</a></li> <li><a href="/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D0%BA%D1%8A%D1%81%D0%BD%D0%B0%D1%82%D0%BE%D1%81%D1%82" title="Непрекъснатост">Непрекъснатост</a></li> <li><a href="/wiki/%D0%9E%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%BE%D1%81%D1%82" class="mw-redirect" title="Определеност">Определеност</a></li> <li><a href="/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D0%B0" title="Производна">Производна</a></li> <li><a href="/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл">Интеграл</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Източници"><span id=".D0.98.D0.B7.D1.82.D0.BE.D1.87.D0.BD.D0.B8.D1.86.D0.B8"></span>Източници</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&veaction=edit&section=17" title="Редактиране на раздел: Източници" class="mw-editsection-visualeditor"><span>редактиране</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F&action=edit&section=17" title="Edit section's source code: Източници"><span>редактиране на кода</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://ibl.bas.bg/rbe/lang/bg/аргумент/">Аргумент в РБЕ, трето значение</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Тодоров, Добромир, Кирил Николов. <i>Математика</i>. <i>Четвърто издание</i>. Стр. 59. УНСС, София, 2009.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Тодоров, Добромир, Кирил Николов. <i>Математика</i>. <i>Четвърто издание</i>. Стр. 60. УНСС, София, 2009.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Тодоров, Добромир, Кирил Николов. <i>Математика</i>. <i>Четвърто издание</i>. Стр. 129. УНСС, София, 2009.</span> </li> </ol></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="">Взето от „<a dir="ltr" href="https://bg.wikipedia.org/w/index.php?title=Функция&oldid=12323192">https://bg.wikipedia.org/w/index.php?title=Функция&oldid=12323192</a>“.</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%B8:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8" title="Специални:Категории">Категория</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8" title="Категория:Математически функции">Математически функции</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Последната промяна на страницата е извършена на 6 август 2024 г. в 15:31 ч.</li> <li id="footer-info-copyright">Текстът е достъпен под лиценза <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.bg">Creative Commons Признание-Споделяне на споделеното</a>; може да са приложени допълнителни условия. 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