Función continua - Wikipedia, a enciclopedia libre

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mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Ferramentas persoais</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú de usuario" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_gl.wikipedia.org&uselang=gl"><span>Doazóns</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crear_unha_conta&returnto=Funci%C3%B3n+continua" title="É recomendable que cree unha conta e acceda ao sistema, se ben non é obrigatorio"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crear unha conta</span></a></li><li id="pt-login" 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class="mw-list-item"><a href="/wiki/Especial:A_mi%C3%B1a_conversa" title="Conversa acerca de edicións feitas desde este enderezo IP [n]" accesskey="n"><span>Conversa</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="Sitio"> <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="Contidos" 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">Contidos</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">agochar</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">Inicio</div> </a> </li> <li id="toc-Historia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historia</span> </div> </a> <ul id="toc-Historia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funcións_reais_dunha_variable_real" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funcións_reais_dunha_variable_real"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Funcións reais dunha variable real</span> </div> </a> <button aria-controls="toc-Funcións_reais_dunha_variable_real-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>Mostrar ou agochar a subsección "Funcións reais dunha variable real"</span> </button> <ul id="toc-Funcións_reais_dunha_variable_real-sublist" class="vector-toc-list"> <li id="toc-Continuidade_dunha_función_nun_punto" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuidade_dunha_función_nun_punto"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Continuidade dunha función nun punto</span> </div> </a> <ul id="toc-Continuidade_dunha_función_nun_punto-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuidade_lateral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuidade_lateral"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Continuidade lateral</span> </div> </a> <ul id="toc-Continuidade_lateral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuidade_dunha_función_nun_intervalo_aberto:_(a,_b)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuidade_dunha_función_nun_intervalo_aberto:_(a,_b)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Continuidade dunha función nun intervalo aberto: (<i>a</i>, <i>b</i>)</span> </div> </a> <ul id="toc-Continuidade_dunha_función_nun_intervalo_aberto:_(a,_b)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuidade_dunha_función_nun_intervalo_pechado:_[a,_b]" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuidade_dunha_función_nun_intervalo_pechado:_[a,_b]"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Continuidade dunha función nun intervalo pechado: [<i>a</i>, <i>b</i>]</span> </div> </a> <ul id="toc-Continuidade_dunha_función_nun_intervalo_pechado:_[a,_b]-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algunhas_funcións_continuas_importantes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algunhas_funcións_continuas_importantes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Algunhas funcións continuas importantes</span> </div> </a> <button aria-controls="toc-Algunhas_funcións_continuas_importantes-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>Mostrar ou agochar a subsección "Algunhas funcións continuas importantes"</span> </button> <ul id="toc-Algunhas_funcións_continuas_importantes-sublist" class="vector-toc-list"> <li id="toc-Funcións_definidas_por_intervalos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Funcións_definidas_por_intervalos"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Funcións definidas por intervalos</span> </div> </a> <ul id="toc-Funcións_definidas_por_intervalos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Función_racional" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Función_racional"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Función racional</span> </div> </a> <ul id="toc-Función_racional-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Teoremas_sobre_funcións_continuas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teoremas_sobre_funcións_continuas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Teoremas sobre funcións continuas</span> </div> </a> <button aria-controls="toc-Teoremas_sobre_funcións_continuas-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>Mostrar ou agochar a subsección "Teoremas sobre funcións continuas"</span> </button> <ul id="toc-Teoremas_sobre_funcións_continuas-sublist" class="vector-toc-list"> <li id="toc-Derivabilidade_e_continuidade" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivabilidade_e_continuidade"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Derivabilidade e continuidade</span> </div> </a> <ul id="toc-Derivabilidade_e_continuidade-sublist" class="vector-toc-list"> <li id="toc-Clase_de_continuidade" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Clase_de_continuidade"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Clase de continuidade</span> </div> </a> <ul id="toc-Clase_de_continuidade-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Funcións_continuas_en_espazos_topolóxicos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funcións_continuas_en_espazos_topolóxicos"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Funcións continuas en espazos topolóxicos</span> </div> </a> <ul id="toc-Funcións_continuas_en_espazos_topolóxicos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funcións_continuas_sobre_os_números_ordinais" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funcións_continuas_sobre_os_números_ordinais"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Funcións continuas sobre os números ordinais</span> </div> </a> <ul id="toc-Funcións_continuas_sobre_os_números_ordinais-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notas"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notas</span> </div> </a> <ul id="toc-Notas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Véxase_tamén" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véxase_tamén"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Véxase tamén</span> </div> </a> <button aria-controls="toc-Véxase_tamén-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>Mostrar ou agochar a subsección "Véxase tamén"</span> </button> <ul id="toc-Véxase_tamén-sublist" class="vector-toc-list"> <li id="toc-Bibliografía" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliografía"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Bibliografía</span> </div> </a> <ul id="toc-Bibliografía-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Outros_artigos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Outros_artigos"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Outros artigos</span> </div> </a> <ul id="toc-Outros_artigos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligazóns_externas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ligazóns_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Ligazóns externas</span> </div> </a> <ul id="toc-Ligazóns_externas-sublist" class="vector-toc-list"> </ul> </li> </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="Contidos" 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="Mostrar ou agochar a táboa de contidos" > <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">Mostrar ou agochar a táboa de contidos</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">Función continua</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="Ir a un artigo noutra lingua. Dispoñible en 58 linguas" > <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-58" 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">58 linguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%AA%E1%8C%8B_%E1%8A%A0%E1%88%B5%E1%88%A8%E1%8A%AB%E1%89%A2" title="ሪጋ አስረካቢ – amhárico" lang="am" hreflang="am" data-title="ሪጋ አስረካቢ" data-language-autonym="አማርኛ" data-language-local-name="amhárico" class="interlanguage-link-target"><span>አማርኛ</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_%D9%85%D8%B3%D8%AA%D9%85%D8%B1%D8%A9" title="دالة مستمرة – árabe" lang="ar" hreflang="ar" data-title="دالة مستمرة" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BF%D0%B5%D1%80%D0%B0%D1%80%D1%8B%D1%9E%D0%BD%D0%B0%D1%8F_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%8B%D1%8F" title="Неперарыўная функцыя – belaruso" lang="be" hreflang="be" data-title="Неперарыўная функцыя" data-language-autonym="Беларуская" data-language-local-name="belaruso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%AC%E0%A6%BF%E0%A6%9A%E0%A7%8D%E0%A6%9B%E0%A6%BF%E0%A6%A8%E0%A7%8D%E0%A6%A8_%E0%A6%AB%E0%A6%BE%E0%A6%82%E0%A6%B6%E0%A6%A8" title="অবিচ্ছিন্ন ফাংশন – bengalí" lang="bn" hreflang="bn" data-title="অবিচ্ছিন্ন ফাংশন" data-language-autonym="বাংলা" data-language-local-name="bengalí" 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/Neprekidna_funkcija" title="Neprekidna funkcija – bosníaco" lang="bs" hreflang="bs" data-title="Neprekidna funkcija" data-language-autonym="Bosanski" data-language-local-name="bosníaco" 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_cont%C3%ADnua" title="Funció contínua – catalán" lang="ca" hreflang="ca" data-title="Funció contínua" data-language-autonym="Català" data-language-local-name="catalán" 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%DB%8C_%D8%A8%DB%95%D8%B1%D8%AF%DB%95%D9%88%D8%A7%D9%85" title="فانکشنی بەردەوام – kurdo central" lang="ckb" hreflang="ckb" data-title="فانکشنی بەردەوام" data-language-autonym="کوردی" data-language-local-name="kurdo central" 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/Spojit%C3%A9_zobrazen%C3%AD" title="Spojité zobrazení – checo" lang="cs" hreflang="cs" data-title="Spojité zobrazení" data-language-autonym="Čeština" data-language-local-name="checo" 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%A2%D0%B0%D1%82%C4%83%D0%BB%D0%BC%D0%B8_%D0%BA%D1%83%C3%A7%D0%B0%D1%80%D1%83" title="Татăлми куçару – chuvaxo" lang="cv" hreflang="cv" data-title="Татăлми куçару" data-language-autonym="Чӑвашла" data-language-local-name="chuvaxo" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Stetige_Funktion" title="Stetige Funktion – alemán" lang="de" hreflang="de" data-title="Stetige Funktion" data-language-autonym="Deutsch" data-language-local-name="alemán" 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%AD%CF%87%CE%B5%CE%B9%CE%B1_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7%CF%82" title="Συνέχεια συνάρτησης – grego" lang="el" hreflang="el" data-title="Συνέχεια συνάρτησης" data-language-autonym="Ελληνικά" data-language-local-name="grego" 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/Continuous_function" title="Continuous function – inglés" lang="en" hreflang="en" data-title="Continuous function" data-language-autonym="English" data-language-local-name="inglés" 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/Kontinua_funkcio" title="Kontinua funkcio – esperanto" lang="eo" hreflang="eo" data-title="Kontinua funkcio" data-language-autonym="Esperanto" data-language-local-name="esperanto" 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_continua" title="Función continua – español" lang="es" hreflang="es" data-title="Función continua" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Funtzio_jarraitu" title="Funtzio jarraitu – éuscaro" lang="eu" hreflang="eu" data-title="Funtzio jarraitu" data-language-autonym="Euskara" data-language-local-name="éuscaro" 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_%D9%BE%DB%8C%D9%88%D8%B3%D8%AA%D9%87" title="تابع پیوسته – persa" lang="fa" hreflang="fa" data-title="تابع پیوسته" data-language-autonym="فارسی" data-language-local-name="persa" 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/Jatkuva_funktio" title="Jatkuva funktio – finés" lang="fi" hreflang="fi" data-title="Jatkuva funktio" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr badge-Q70893996 mw-list-item" title=""><a href="https://fr.wikipedia.org/wiki/Fonction_continue" title="Fonction continue – francés" lang="fr" hreflang="fr" data-title="Fonction continue" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</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_%D7%A8%D7%A6%D7%99%D7%A4%D7%94_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94)" title="פונקציה רציפה (אנליזה) – hebreo" lang="he" hreflang="he" data-title="פונקציה רציפה (אנליזה)" data-language-autonym="עברית" data-language-local-name="hebreo" 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%B8%E0%A4%A4%E0%A4%A4_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="सतत फलन – hindi" lang="hi" hreflang="hi" data-title="सतत फलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Neprekidnost_funkcije" title="Neprekidnost funkcije – croata" lang="hr" hreflang="hr" data-title="Neprekidnost funkcije" data-language-autonym="Hrvatski" data-language-local-name="croata" 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/Folytonos_f%C3%BCggv%C3%A9ny" title="Folytonos függvény – húngaro" lang="hu" hreflang="hu" data-title="Folytonos függvény" data-language-autonym="Magyar" data-language-local-name="húngaro" 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/%D4%B1%D5%B6%D5%A8%D5%B6%D5%A4%D5%B0%D5%A1%D5%BF_%D5%A1%D6%80%D5%BF%D5%A1%D5%BA%D5%A1%D5%BF%D5%AF%D5%A5%D6%80%D5%B8%D6%82%D5%B4" title="Անընդհատ արտապատկերում – armenio" lang="hy" hreflang="hy" data-title="Անընդհատ արտապատկերում" data-language-autonym="Հայերեն" data-language-local-name="armenio" 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_continue" title="Function continue – interlingua" lang="ia" hreflang="ia" data-title="Function continue" data-language-autonym="Interlingua" data-language-local-name="interlingua" 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_kontinu" title="Fungsi kontinu – indonesio" lang="id" hreflang="id" data-title="Fungsi kontinu" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_continua" title="Funzione continua – italiano" lang="it" hreflang="it" data-title="Funzione continua" data-language-autonym="Italiano" data-language-local-name="italiano" 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%80%A3%E7%B6%9A%E5%86%99%E5%83%8F" title="連続写像 – xaponés" lang="ja" hreflang="ja" data-title="連続写像" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A3%E1%83%AC%E1%83%A7%E1%83%95%E1%83%94%E1%83%A2%E1%83%9D%E1%83%91%E1%83%90" title="უწყვეტობა – xeorxiano" lang="ka" hreflang="ka" data-title="უწყვეტობა" data-language-autonym="ქართული" data-language-local-name="xeorxiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%AE%D0%B7%D1%96%D0%BB%D1%96%D1%81%D1%81%D1%96%D0%B7_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F" title="Үзіліссіз функция – kazako" lang="kk" hreflang="kk" data-title="Үзіліссіз функция" data-language-autonym="Қазақша" data-language-local-name="kazako" 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/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수 – coreano" lang="ko" hreflang="ko" data-title="연속 함수" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tolydi_funkcija" title="Tolydi funkcija – lituano" lang="lt" hreflang="lt" data-title="Tolydi funkcija" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Fungsi_selanjar" title="Fungsi selanjar – malaio" lang="ms" hreflang="ms" data-title="Fungsi selanjar" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Continue_functie_(analyse)" title="Continue functie (analyse) – neerlandés" lang="nl" hreflang="nl" data-title="Continue functie (analyse)" data-language-autonym="Nederlands" data-language-local-name="neerlandés" 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/Kontinuerleg_funksjon" title="Kontinuerleg funksjon – noruegués nynorsk" lang="nn" hreflang="nn" data-title="Kontinuerleg funksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegués nynorsk" 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/Kontinuerlig_funksjon" title="Kontinuerlig funksjon – noruegués bokmål" lang="nb" hreflang="nb" data-title="Kontinuerlig funksjon" data-language-autonym="Norsk bokmål" data-language-local-name="noruegués bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A8%E0%A8%BF%E0%A8%B0%E0%A9%B0%E0%A8%A4%E0%A8%B0_%E0%A8%AB%E0%A9%B0%E0%A8%95%E0%A8%B8%E0%A8%BC%E0%A8%A8" title="ਨਿਰੰਤਰ ਫੰਕਸ਼ਨ – panxabí" lang="pa" hreflang="pa" data-title="ਨਿਰੰਤਰ ਫੰਕਸ਼ਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panxabí" 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_ci%C4%85g%C5%82a" title="Funkcja ciągła – polaco" lang="pl" hreflang="pl" data-title="Funkcja ciągła" data-language-autonym="Polski" data-language-local-name="polaco" 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_continua" title="Fonsion continua – Piedmontese" lang="pms" hreflang="pms" data-title="Fonsion continua" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_cont%C3%ADnua" title="Função contínua – portugués" lang="pt" hreflang="pt" data-title="Função contínua" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Func%C8%9Bie_continu%C4%83" title="Funcție continuă – romanés" lang="ro" hreflang="ro" data-title="Funcție continuă" data-language-autonym="Română" data-language-local-name="romanés" 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%9D%D0%B5%D0%BF%D1%80%D0%B5%D1%80%D1%8B%D0%B2%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Непрерывное отображение – ruso" lang="ru" hreflang="ru" data-title="Непрерывное отображение" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Neprekidne_funkcije" title="Neprekidne funkcije – serbocroata" lang="sh" hreflang="sh" data-title="Neprekidne funkcije" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" 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/Continuous_function" title="Continuous function – Simple English" lang="en-simple" hreflang="en-simple" data-title="Continuous function" 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/Spojit%C3%A1_funkcia" title="Spojitá funkcia – eslovaco" lang="sk" hreflang="sk" data-title="Spojitá funkcia" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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/Zvezna_funkcija" title="Zvezna funkcija – esloveno" lang="sl" hreflang="sl" data-title="Zvezna funkcija" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D0%BA%D0%B8%D0%B4%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Непрекидна функција – serbio" lang="sr" hreflang="sr" data-title="Непрекидна функција" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kontinuerlig_funktion" title="Kontinuerlig funktion – sueco" lang="sv" hreflang="sv" data-title="Kontinuerlig funktion" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8A%E0%AE%9F%E0%AE%B0%E0%AF%8D%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AE%BF%E0%AE%AF%E0%AE%BE%E0%AE%A9_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="தொடர்ச்சியான சார்பு – támil" lang="ta" hreflang="ta" data-title="தொடர்ச்சியான சார்பு" data-language-autonym="தமிழ்" data-language-local-name="támil" 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%95%E0%B9%88%E0%B8%AD%E0%B9%80%E0%B8%99%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%87" title="ฟังก์ชันต่อเนื่อง – tailandés" lang="th" hreflang="th" data-title="ฟังก์ชันต่อเนื่อง" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BF%D0%B5%D1%80%D0%B5%D1%80%D0%B2%D0%BD%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F" title="Неперервна функція – ucraíno" lang="uk" hreflang="uk" data-title="Неперервна функція" data-language-autonym="Українська" data-language-local-name="ucraíno" 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%A7%D8%B3%D8%AA%D9%85%D8%B1%DB%8C_%D8%AF%D8%A7%D9%84%DB%81" title="استمری دالہ – urdú" lang="ur" hreflang="ur" data-title="استمری دالہ" data-language-autonym="اردو" data-language-local-name="urdú" 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/Uzluksiz_funksiya" title="Uzluksiz funksiya – uzbeko" lang="uz" hreflang="uz" data-title="Uzluksiz funksiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeko" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_li%C3%AAn_t%E1%BB%A5c" title="Hàm liên tục – vietnamita" lang="vi" hreflang="vi" data-title="Hàm liên tục" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0" title="连续函数 – chinés wu" lang="wuu" hreflang="wuu" data-title="连续函数" data-language-autonym="吴语" data-language-local-name="chinés wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%A3%E1%83%AD%E1%83%A7%E1%83%95%E1%83%90%E1%83%93%E1%83%A3_%E1%83%A4%E1%83%A3%E1%83%9C%E1%83%A5%E1%83%AA%E1%83%98%E1%83%90" title="უჭყვადუ ფუნქცია – Mingrelian" lang="xmf" hreflang="xmf" data-title="უჭყვადუ ფუნქცია" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0" title="连续函数 – chinés" lang="zh" hreflang="zh" data-title="连续函数" data-language-autonym="中文" data-language-local-name="chinés" 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/%E9%80%A3%E7%BA%8C" title="連續 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="連續" 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-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%80%A3%E7%BA%8C%E5%87%BD%E6%95%B8" title="連續函數 – cantonés" lang="yue" hreflang="yue" data-title="連續函數" data-language-autonym="粵語" data-language-local-name="cantonés" 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/Q170058#sitelinks-wikipedia" title="Editar as ligazóns interlingüísticas" class="wbc-editpage">Editar as ligazóns</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="Espazos de nomes"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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</nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Na Galipedia, a Wikipedia en galego.</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="gl" dir="ltr"><p>En <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a>, unha <b>función continua</b> é aquela para a que, intuitivamente, para puntos próximos do dominio prodúcense pequenas variacións nos valores da función; aínda que en rigor, nun espazo métrico significa o contrario, que pequenas variacións da <b><a href="/wiki/Funci%C3%B3n" title="Función">función</a></b> implican que deben estar próximos os puntos. Se a función non é continua, dise que é <b>descontinua</b>. Unha función continua de <span class="mwe-math-element"><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> en <span class="mwe-math-element"><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> é aquela cunha <a href="/wiki/Gr%C3%A1fica_dunha_funci%C3%B3n" title="Gráfica dunha función">gráfica</a> que pode debuxarse sen levantar o lapis do papel (máis formalmente a súa gráfica é un <a href="/wiki/Conxunto_conexo" title="Conxunto conexo">conxunto conexo</a>). </p><p>A continuidade de funcións é un dos conceptos principais da <a href="/wiki/An%C3%A1lise_matem%C3%A1tica" title="Análise matemática">análise matemática</a> e da <a href="/wiki/Topolox%C3%ADa" title="Topoloxía">topoloxía</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=1" title="Editar a sección: «Historia»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=1" title="Editar o código fonte da sección: Historia"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unha primeira forma da definición (ε, δ) de continuidade foi dada por <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a> en 1817. <a href="/wiki/Augustin-Louis_Cauchy" class="mw-redirect" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> definiu a continuidade de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span> como segue: un incremento infinitamente pequeno <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> da variable independente <i>x</i> sempre produce un cambio infinitamente pequeno <span class="mwe-math-element"><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+\alpha )-f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+\alpha )-f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cdc7a8529de4a0492b62de55ca5c0bc15ca2651" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.004ex; height:2.843ex;" alt="{\displaystyle f(x+\alpha )-f(x)}"></span> da variable dependente <i>y</i>. Cauchy definiu cantitades infinitamente pequenas en termos de cantidades variables, e a súa definición de continuidade é paralela á definición infinitesimal empregada na actualidade. A definición e a distinción entre continuidade nun punto e <a href="/wiki/Continuidade_uniforme" title="Continuidade uniforme">continuidade uniforme</a> foi dada por primeira vez por Bolzano na década de 1830 mais a súa obra non foi publicada ata cen anos despois. Como Bolzano,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> negou a continuidade dunha función nun punto <i>c</i> a menos que estivese definida a ambos os lados de <i>c</i>, mais <a href="/w/index.php?title=%C3%89douard_Goursat&action=edit&redlink=1" class="new" title="Édouard Goursat (a páxina aínda non existe)">Édouard Goursat</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>3<span>]</span></a></sup> permitía que unha función estivese definida só nun lado de <i>c</i>, e <a href="/wiki/Camille_Jordan" title="Camille Jordan">Camille Jordan</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>4<span>]</span></a></sup> incluso se a función estaba definida só no punto <i>c</i>. Todas estas definicións non equivalentes de continuidade nun punto aínda se empregan.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>5<span>]</span></a></sup> </p><p><a href="/w/index.php?title=Eduard_Heine&action=edit&redlink=1" class="new" title="Eduard Heine (a páxina aínda non existe)">Eduard Heine</a> achegou a primeira definición de continuidade uniforme en 1872, pero baseou estas ideas en traballos de <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a> de 1854.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span>[</span>6<span>]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Funcións_reais_dunha_variable_real"><span id="Funci.C3.B3ns_reais_dunha_variable_real"></span>Funcións reais dunha variable real</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=2" title="Editar a sección: «Funcións reais dunha variable real»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=2" title="Editar o código fonte da sección: Funcións reais dunha variable real"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Ficheiro:Funci%C3%B3n_Continua_011.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Funci%C3%B3n_Continua_011.svg/300px-Funci%C3%B3n_Continua_011.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Funci%C3%B3n_Continua_011.svg/450px-Funci%C3%B3n_Continua_011.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Funci%C3%B3n_Continua_011.svg/600px-Funci%C3%B3n_Continua_011.svg.png 2x" data-file-width="1000" data-file-height="800" /></a><figcaption></figcaption></figure> <p>Informalmente falando, unha función <i>f</i> definida sobre un <a href="/wiki/Intervalo_(matem%C3%A1ticas)" title="Intervalo (matemáticas)">intervalo</a> <b>I</b> é continua se a curva que a representa, é dicir o conxunto dos puntos (<i>x</i>, <i>f</i>(<i>x</i>)), con <i>x</i> en <b>I</b>, está constituída por un trazo continuo, é dicir un trazo que non está roto, nin ten "ocos" nin "saltos", como na figura da dereita. </p><p>O intervalo <b>I</b> de <i>x</i> é o <a href="/wiki/Dominio_de_definici%C3%B3n" title="Dominio de definición">dominio de definición</a> de <i>f</i>, definido como o conxunto dos valores de <i>x</i> para os cales existe <i>f</i>(<i>x</i>). </p><p>O intervalo <b>J</b> de <i>y</i> é o rango (tamén coñecido como imaxe) de <i>f</i>, o conxunto dos valores de <i>y</i>, tomados como <i>y</i> = <i>f</i>(<i>x</i>). Escríbese <b>J</b> = <i>f</i>(<b>I</b>). Notar que en xeral, non é igual que o <a href="/wiki/Codominio" title="Codominio">codominio</a> (só é igual se a función é <a href="/wiki/Sobrexectiva" class="mw-redirect" title="Sobrexectiva">sobrexectiva</a>.) </p><p>O maior elemento de <b>J</b> chámase o máximo absoluto de <i>f</i> en <b>I</b>, e o menor valor de <b>J</b> é o seu mínimo absoluto no dominio <b>I</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Continuidade_dunha_función_nun_punto"><span id="Continuidade_dunha_funci.C3.B3n_nun_punto"></span>Continuidade dunha función nun punto</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=3" title="Editar a sección: «Continuidade dunha función nun punto»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=3" title="Editar o código fonte da sección: Continuidade dunha función nun punto"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Ficheiro:Funci%C3%B3n_Continua_014.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Funci%C3%B3n_Continua_014.svg/300px-Funci%C3%B3n_Continua_014.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Funci%C3%B3n_Continua_014.svg/450px-Funci%C3%B3n_Continua_014.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Funci%C3%B3n_Continua_014.svg/600px-Funci%C3%B3n_Continua_014.svg.png 2x" data-file-width="1000" data-file-height="800" /></a><figcaption></figcaption></figure> <p>Unha función <i>f</i> é continua nun punto <i>x</i><sub>0</sub> no dominio da función se: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \varepsilon >0\quad \exists \delta >0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mspace width="1em" /> <mi mathvariant="normal">∃</mi> <mi>δ</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\quad \exists \delta >0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73449d57a9317f79d3dc7d199d0869b995763257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.207ex; height:2.343ex;" alt="{\displaystyle \forall \varepsilon >0\quad \exists \delta >0\;}"></span></dd></dl> <p>tal que para toda <i>x</i> no dominio da función: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-x_{0}|<\delta \quad \Rightarrow \quad |f(x)-f(x_{0})|<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>δ</mi> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </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 class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x-x_{0}|<\delta \quad \Rightarrow \quad |f(x)-f(x_{0})|<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9ccdac0ca947d6ff348b013f041d623bfa33cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.459ex; height:2.843ex;" alt="{\displaystyle |x-x_{0}|<\delta \quad \Rightarrow \quad |f(x)-f(x_{0})|<\varepsilon }"></span></dd></dl> <p>Isto pódese escribir en termos de límites da seguinte maneira: Se <i>x</i><sub>0</sub> é <a href="/w/index.php?title=Punto_de_acumulaci%C3%B3n&action=edit&redlink=1" class="new" title="Punto de acumulación (a páxina aínda non existe)">punto de acumulación</a> do dominio da función entón é continua en <i>x</i><sub>0</sub> se e só se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to x_{0}}f(x)=f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to x_{0}}f(x)=f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6deb947478c24b1d730b1dcf858aeb06ba1249" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.73ex; height:4.176ex;" alt="{\displaystyle \lim _{x\to x_{0}}f(x)=f(x_{0})}"></span>. Cando <i>x</i><sub>0</sub> non é de acumulación do dominio, a función é continua nese punto. </p><p>No caso das aplicacións de <span class="mwe-math-element"><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> en <span class="mwe-math-element"><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>, e dunha maneira máis rigorosa dise que unha función <i>f</i> é continua nun punto <i>x</i><sub>1</sub> se existe <i>f</i>(<i>x</i>1), se existe o límite de <i>f</i>(<i>x</i>) cando <i>x</i> tende a <i>x</i><sub>1</sub> pola dereita, se existe o <a href="/wiki/L%C3%ADmite_matem%C3%A1tico" title="Límite matemático">límite</a> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0" /> </mrow> <annotation encoding="application/x-tex">{\displaystyle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dcd61276328f7c7ec5bdc399b6e11114a2c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0; height:0.343ex;" alt="{\displaystyle }"></span> (x) cando x tende a <i>x</i><sub>1</sub> pola esquerda, e ademais ambos coinciden con <i>f</i>(<i>x</i><sub>1</sub>).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span>[</span>7<span>]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {Blue}(7)}\;f(x_{1})=L_{(x_{1})}\left\{{\begin{array}{l}{\color {Blue}(5)}\;L_{(x_{1})}=L_{(x_{1})}^{+}=L_{(x_{1})}^{-}\left\{{\begin{array}{l}{\color {Blue}(3)}\;\exists \;L_{(x_{1})}^{+}\land \exists \;L_{(x_{1})}^{-}\;\left\{{\begin{array}{l}{\color {Blue}(1)}\;\exists \;L_{(x_{1})}^{+}={\displaystyle \lim _{x\to {x_{1}}^{+}}f(x)}\\\\{\color {Blue}(2)}\;\exists \;L_{(x_{1})}^{-}={\displaystyle \lim _{x\to {x_{1}}^{-}}f(x)}\end{array}}\right.\\{\color {Blue}(4)}\;L_{(x_{1})}^{+}=L_{(x_{1})}^{-}\end{array}}\right.\\{\color {Blue}(6)}\;\exists f(x_{1})\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <mspace width="thickmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <mspace width="thickmathspace" /> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mspace width="thickmathspace" /> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>∧</mo> <mi mathvariant="normal">∃</mi> <mspace width="thickmathspace" /> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mspace width="thickmathspace" /> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mspace width="thickmathspace" /> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mspace width="thickmathspace" /> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <mspace width="thickmathspace" /> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#2D2F92"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {Blue}(7)}\;f(x_{1})=L_{(x_{1})}\left\{{\begin{array}{l}{\color {Blue}(5)}\;L_{(x_{1})}=L_{(x_{1})}^{+}=L_{(x_{1})}^{-}\left\{{\begin{array}{l}{\color {Blue}(3)}\;\exists \;L_{(x_{1})}^{+}\land \exists \;L_{(x_{1})}^{-}\;\left\{{\begin{array}{l}{\color {Blue}(1)}\;\exists \;L_{(x_{1})}^{+}={\displaystyle \lim _{x\to {x_{1}}^{+}}f(x)}\\\\{\color {Blue}(2)}\;\exists \;L_{(x_{1})}^{-}={\displaystyle \lim _{x\to {x_{1}}^{-}}f(x)}\end{array}}\right.\\{\color {Blue}(4)}\;L_{(x_{1})}^{+}=L_{(x_{1})}^{-}\end{array}}\right.\\{\color {Blue}(6)}\;\exists f(x_{1})\end{array}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/812f46cfa6583d73aa92cf270fe6560f8ee941b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:95.414ex; height:20.509ex;" alt="{\displaystyle {\color {Blue}(7)}\;f(x_{1})=L_{(x_{1})}\left\{{\begin{array}{l}{\color {Blue}(5)}\;L_{(x_{1})}=L_{(x_{1})}^{+}=L_{(x_{1})}^{-}\left\{{\begin{array}{l}{\color {Blue}(3)}\;\exists \;L_{(x_{1})}^{+}\land \exists \;L_{(x_{1})}^{-}\;\left\{{\begin{array}{l}{\color {Blue}(1)}\;\exists \;L_{(x_{1})}^{+}={\displaystyle \lim _{x\to {x_{1}}^{+}}f(x)}\\\\{\color {Blue}(2)}\;\exists \;L_{(x_{1})}^{-}={\displaystyle \lim _{x\to {x_{1}}^{-}}f(x)}\end{array}}\right.\\{\color {Blue}(4)}\;L_{(x_{1})}^{+}=L_{(x_{1})}^{-}\end{array}}\right.\\{\color {Blue}(6)}\;\exists f(x_{1})\end{array}}\right.}"></span></dd></dl> <p>Así pois, unha función <i>f</i> continua no punto <i>x</i><sub>1</sub> implica o seguinte: 1. Existe o límite pola dereita: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \lim _{x\to x_{1}^{+}}f(x)\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</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 \exists \lim _{x\to x_{1}^{+}}f(x)\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d27addf9ec28d50c557e9bc59478708934c0a179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.729ex; height:5.009ex;" alt="{\displaystyle \exists \lim _{x\to x_{1}^{+}}f(x)\in \mathbb {R} }"></span></dd></dl> <p>2. Existe o límite pola esquerda: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \lim _{x\to x_{1}^{-}}f(x)\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</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 \exists \lim _{x\to x_{1}^{-}}f(x)\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd23785f20adfe4f40e0034945f02007d4555ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.729ex; height:5.009ex;" alt="{\displaystyle \exists \lim _{x\to x_{1}^{-}}f(x)\in \mathbb {R} }"></span></dd></dl> <p>3. A función ten límite pola dereita e pola esquerda do punto <i>x</i><sub>1</sub> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \lim _{x\to x_{1}^{+}}f(x)\in \mathbb {R} \quad \land \quad \exists \lim _{x\to x_{1}^{-}}f(x)\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mspace width="1em" /> <mo>∧</mo> <mspace width="1em" /> <mi mathvariant="normal">∃</mi> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</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 \exists \lim _{x\to x_{1}^{+}}f(x)\in \mathbb {R} \quad \land \quad \exists \lim _{x\to x_{1}^{-}}f(x)\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18fa7f11d2848526ba417ecd80ecd8edbe4dd99a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.685ex; height:5.009ex;" alt="{\displaystyle \exists \lim _{x\to x_{1}^{+}}f(x)\in \mathbb {R} \quad \land \quad \exists \lim _{x\to x_{1}^{-}}f(x)\in \mathbb {R} }"></span></dd></dl> <p>4. O límite pola dereita e o límite pola esquerda coinciden: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <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> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0af6d2f0d2b9d3cab7a960fca26f2b1efac2cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.159ex; height:5.009ex;" alt="{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)}"></span></dd></dl> <p>5. Se existen o límite pola dereita e pola esquerda e os seus valores coinciden, a función ten límite neste punto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to x_{1}}f(x)=\lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>1</mn> </mrow> </msub> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <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> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <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> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to x_{1}}f(x)=\lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b49ab216e70414fb998c51c12b3d3e3276d3778c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.418ex; height:5.009ex;" alt="{\displaystyle \lim _{x\to x_{1}}f(x)=\lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)}"></span></dd></dl> <p>6. Existe <i>f</i>(<i>x</i><sub>1</sub>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists f(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists f(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce06f2a873dc12429424157a3153fea4544a2f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.764ex; height:2.843ex;" alt="{\displaystyle \exists f(x_{1})}"></span></dd></dl> <p>7. O límite e o valor da función coinciden: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to x_{1}}f(x)=f(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>1</mn> </mrow> </msub> </mrow> </munder> <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>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to x_{1}}f(x)=f(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9c71599d6a7fd7ea01b0e7cb4970c189c65b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.73ex; height:4.176ex;" alt="{\displaystyle \lim _{x\to x_{1}}f(x)=f(x_{1})}"></span></dd></dl> <p>A función é continua nese punto. Unha función é continua nun intervalo se é continua en todos os seus puntos. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Ficheiro:Funci%C3%B3n_Continua_022.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Funci%C3%B3n_Continua_022.svg/300px-Funci%C3%B3n_Continua_022.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Funci%C3%B3n_Continua_022.svg/450px-Funci%C3%B3n_Continua_022.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Funci%C3%B3n_Continua_022.svg/600px-Funci%C3%B3n_Continua_022.svg.png 2x" data-file-width="1000" data-file-height="800" /></a><figcaption></figcaption></figure> <p>Se <i>f</i>(<i>x</i><sub>1</sub>)= <i>y</i><sub>1</sub>, a continuidade en <i>x</i><sub>1</sub> exprésase así: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to x_{1}}f(x)=y_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>1</mn> </mrow> </msub> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to x_{1}}f(x)=y_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a93cbcf8714c3671854dcfba35779a386b1a7eda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.452ex; height:4.176ex;" alt="{\displaystyle \lim _{x\to x_{1}}f(x)=y_{1}}"></span></dd></dl> <p>Parafraseando, cando <b>x</b> se aproxima a <i>x</i><sub>1</sub>, <i>f</i>(<i>x</i>) aproxímase a <i>y</i><sub>1</sub>. Por definición dos límites, isto significa que para todo intervalo aberto <b>J</b>, centrado en <i>y</i><sub>1</sub>, existe un intervalo aberto <b>I</b>, centrado en <i>x</i><sub>1</sub>, tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(I)\in J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(I)\in J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d3fbc8a318a159d82b2a48b73e2461da66eddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.572ex; height:2.843ex;" alt="{\displaystyle f(I)\in J}"></span>. </p><p>Se <i>f</i> ten un salto no punto, o teorema non se cumpre. En efecto, non todo intervalo <b>I</b> ao redor de <i>x</i><sub>1</sub> ten a súa imaxe nun intervalo <b>J</b> centrado en <i>y</i><sub>1</sub>, cun raio inferior ao salto de <i>f</i>; non importa o pequeno que este intervalo sexa, hai valores de <i>x</i> do intervalo <b>I</b> ao redor de <i>x</i><sub>1</sub> que ten a súa imaxe nun intervalo <b>K</b> centrado en <i>y</i><sub>2</sub>, sendo <i>y</i><sub>1</sub> e <i>y</i><sub>2</sub> valores distintos, isto é: <i>x</i> ten imaxes que saen de <b>J</b>. </p><p>A vantaxe desta definición é que se xeneraliza a calquera <a href="/wiki/Espazo_topol%C3%B3xico" title="Espazo topolóxico">espazo topolóxico</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Continuidade_lateral">Continuidade lateral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=4" title="Editar a sección: «Continuidade lateral»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=4" title="Editar o código fonte da sección: Continuidade lateral"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Ficheiro:Funci%C3%B3n_Continua_024.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Funci%C3%B3n_Continua_024.svg/300px-Funci%C3%B3n_Continua_024.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Funci%C3%B3n_Continua_024.svg/450px-Funci%C3%B3n_Continua_024.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Funci%C3%B3n_Continua_024.svg/600px-Funci%C3%B3n_Continua_024.svg.png 2x" data-file-width="1000" data-file-height="800" /></a><figcaption></figcaption></figure> <p>Unha función <i>f</i> é <b>continua pola esquerda</b> no punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdf4f9973d3d0c15948ee465d462b13151563d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{1}}"></span> se o límite lateral pola esquerda e o valor da función no punto son iguais. É dicir: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=f(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </munder> <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>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=f(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec6b07a8becb5172c1137c683a2501a6c7f5812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.101ex; height:5.009ex;" alt="{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=f(x_{1})}"></span></dd></dl> <p>como na figura. </p><p>Unha función <i>f</i> é <b>continua pola dereita</b> no punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdf4f9973d3d0c15948ee465d462b13151563d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle x=x_{1}}"></span> se o seu límite lateral pola dereita e o valor da función no punto son iguais. É dicir: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to x_{1}^{+}}f(x)=f(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> </munder> <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>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to x_{1}^{+}}f(x)=f(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d71639702e47213e1cf06f33ba03593f7f152f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.101ex; height:5.009ex;" alt="{\displaystyle \lim _{x\to x_{1}^{+}}f(x)=f(x_{1})}"></span></dd></dl> <p>Unha función <i>f</i> é <b>continua nun punto</b> se é <b>continua pola esquerda</b> e é <b>continua pola dereita.</b> Isto é: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)=f(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <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> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> </munder> <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>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)=f(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bfe0f7ac0e55a81edd3689fb6b6d5d378f5c622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.73ex; height:5.009ex;" alt="{\displaystyle \lim _{x\to x_{1}^{-}}f(x)=\lim _{x\to x_{1}^{+}}f(x)=f(x_{1})}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Continuidade_dunha_función_nun_intervalo_aberto:_(a,_b)"><span id="Continuidade_dunha_funci.C3.B3n_nun_intervalo_aberto:_.28a.2C_b.29"></span>Continuidade dunha función nun intervalo aberto: (<i>a</i>, <i>b</i>)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=5" title="Editar a sección: «Continuidade dunha función nun intervalo aberto: (a, b)»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=5" title="Editar o código fonte da sección: Continuidade dunha función nun intervalo aberto: (a, b)"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un valor <i>c</i> pertence a un <a href="/wiki/Intervalo_(matem%C3%A1ticas)" title="Intervalo (matemáticas)">intervalo</a> aberto <b>I</b>, de extremo esquerdo <i>a</i> e extremo dereito <i>b</i>, representado <b>I</b>= (<i>a</i>, <i>b</i>) se: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<c<b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>c</mi> <mo><</mo> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<c<b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb50bfe26256fa0bf6f0defda78984f98ce9985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.076ex; height:2.176ex;" alt="{\displaystyle a<c<b\;}"></span></dd></dl> <p>Unha función <i>f</i> é continua nun intervalo aberto <b>I</b>= (<i>a</i>, <i>b</i>), se e só se a función é continua en todos os puntos do intervalo, é dicir: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall c\in I=(a,b):\quad \lim _{x\to c}f(x)=f(c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>c</mi> <mo>∈</mo> <mi>I</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall c\in I=(a,b):\quad \lim _{x\to c}f(x)=f(c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1830e68943b6774418ec0d98a0c6335ae73edf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.034ex; height:3.843ex;" alt="{\displaystyle \forall c\in I=(a,b):\quad \lim _{x\to c}f(x)=f(c)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Continuidade_dunha_función_nun_intervalo_pechado:_[a,_b]"><span id="Continuidade_dunha_funci.C3.B3n_nun_intervalo_pechado:_.5Ba.2C_b.5D"></span>Continuidade dunha función nun intervalo pechado: [<i>a</i>, <i>b</i>]</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=6" title="Editar a sección: «Continuidade dunha función nun intervalo pechado: [a, b]»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=6" title="Editar o código fonte da sección: Continuidade dunha función nun intervalo pechado: [a, b]"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un valor <i>c</i> pertence a un intervalo pechado <b>I</b>, de extremo esquerdo <i>a</i> e extremo dereito <i>b</i>, representado <b>I</b>= [<i>a</i>, <i>b</i>] se: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq c\leq b\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <mi>c</mi> <mo>≤</mo> <mi>b</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq c\leq b\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb68cf9c308dd20f656ad592b8a16e1030663cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.076ex; height:2.343ex;" alt="{\displaystyle a\leq c\leq b\;}"></span></dd></dl> <p>Unha función <i>f</i> é continua nun intervalo pechado [<i>a</i>, <i>b</i>] se a función é continua no intervalo aberto (<i>a</i>, <i>b</i>) e é continua pola dereita da e continua pola esquerda de <i>b</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall c\in I=[a,b]:\quad \lim _{x\to c}f(x)=f(c)\quad \land \quad \lim _{x\to a^{+}}f(x)=f(a)\quad \land \quad \lim _{x\to b^{-}}f(x)=f(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>c</mi> <mo>∈</mo> <mi>I</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>:</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>c</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo>∧</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo>∧</mo> <mspace width="1em" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall c\in I=[a,b]:\quad \lim _{x\to c}f(x)=f(c)\quad \land \quad \lim _{x\to a^{+}}f(x)=f(a)\quad \land \quad \lim _{x\to b^{-}}f(x)=f(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/492ec2b396809c01662d6c821de2db488fdd1be3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:81.329ex; height:4.343ex;" alt="{\displaystyle \forall c\in I=[a,b]:\quad \lim _{x\to c}f(x)=f(c)\quad \land \quad \lim _{x\to a^{+}}f(x)=f(a)\quad \land \quad \lim _{x\to b^{-}}f(x)=f(b)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Algunhas_funcións_continuas_importantes"><span id="Algunhas_funci.C3.B3ns_continuas_importantes"></span>Algunhas funcións continuas importantes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=7" title="Editar a sección: «Algunhas funcións continuas importantes»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=7" title="Editar o código fonte da sección: Algunhas funcións continuas importantes"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Funci%C3%B3n_Trigonom%C3%A9trica_R110.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Funci%C3%B3n_Trigonom%C3%A9trica_R110.svg/400px-Funci%C3%B3n_Trigonom%C3%A9trica_R110.svg.png" decoding="async" width="400" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Funci%C3%B3n_Trigonom%C3%A9trica_R110.svg/600px-Funci%C3%B3n_Trigonom%C3%A9trica_R110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Funci%C3%B3n_Trigonom%C3%A9trica_R110.svg/800px-Funci%C3%B3n_Trigonom%C3%A9trica_R110.svg.png 2x" data-file-width="1000" data-file-height="700" /></a><figcaption>Funcións seno e coseno.</figcaption></figure> <p>As funcións <a href="/wiki/Polinomio" title="Polinomio">polinómicas</a>, <a href="/wiki/Trigonometr%C3%ADa" title="Trigonometría">trigonométricas</a>: <a href="/wiki/Seno" title="Seno">seno</a> e <a href="/wiki/Coseno" title="Coseno">coseno</a>, as <a href="/wiki/Funci%C3%B3n_exponencial" title="Función exponencial">exponenciais</a> e as <a href="/wiki/Logaritmo" title="Logaritmo">logarítmicas</a> son continuas nos seus respectivos dominios de definición. </p><p>A <a href="/wiki/Par%C3%A1bola_(xeometr%C3%ADa)" title="Parábola (xeometría)">parábola</a>, como función polinómica, é un exemplo de función continua ao longo de todo o dominio real. </p><p>Na gráfica vese a función seno que é periódica, limitada e continua en todo o domino real. Dado o seu carácter periódico, con ver un só dos ciclos é suficiente para comprobar a continuidade, porque o resto dos ciclos son exactamente iguais. </p> <div class="mw-heading mw-heading3"><h3 id="Funcións_definidas_por_intervalos"><span id="Funci.C3.B3ns_definidas_por_intervalos"></span>Funcións definidas por intervalos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=8" title="Editar a sección: «Funcións definidas por intervalos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=8" title="Editar o código fonte da sección: Funcións definidas por intervalos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Ficheiro:Funci%C3%B3n_Continua_050.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Funci%C3%B3n_Continua_050.svg/300px-Funci%C3%B3n_Continua_050.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Funci%C3%B3n_Continua_050.svg/450px-Funci%C3%B3n_Continua_050.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Funci%C3%B3n_Continua_050.svg/600px-Funci%C3%B3n_Continua_050.svg.png 2x" data-file-width="1000" data-file-height="800" /></a><figcaption></figcaption></figure> <p>As funcións definidas para distintos intervalos de <i>x</i> poden ser descontinuas nos puntos de cambio de intervalo, por exemplo: </p> <ul><li>A <a href="/w/index.php?title=Funci%C3%B3n_parte_enteira&action=edit&redlink=1" class="new" title="Función parte enteira (a páxina aínda non existe)">función parte enteira</a> de <i>x</i>, <i>E</i>(<i>x</i>), onde <i>E</i>(<i>x</i>) é o maior <a href="/wiki/N%C3%BAmero_enteiro" title="Número enteiro">número enteiro</a> inferior ou igual a <i>x</i>, tal que:</li></ul> <dl><dd>E(<i>x</i>) ≤ <i>x</i> < E(<i>x</i>) + 1.</dd></dl> <p>A súa gráfica é unha sucesión de segmentos horizontais a distintas alturas. Esta función non é continua nos enteiros, pois os límites á esquerda e á dereita difiren dun, pero é continua nos segmentos abertos (<i>n</i>, <i>n</i>+1) onde é constante. </p> <ul><li>Outra función definida por intervalo é a <a href="/wiki/Funci%C3%B3n_signo" title="Función signo">función signo</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Función_racional"><span id="Funci.C3.B3n_racional"></span>Función racional</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=9" title="Editar a sección: «Función racional»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=9" title="Editar o código fonte da sección: Función racional"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/Ficheiro:Funci%C3%B3n_Continua_033.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Funci%C3%B3n_Continua_033.svg/300px-Funci%C3%B3n_Continua_033.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Funci%C3%B3n_Continua_033.svg/450px-Funci%C3%B3n_Continua_033.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Funci%C3%B3n_Continua_033.svg/600px-Funci%C3%B3n_Continua_033.svg.png 2x" data-file-width="1000" data-file-height="800" /></a><figcaption></figcaption></figure> <p>As funcións racionais son continuas nun intervalo axeitado. Un exemplo disto é a función inverso de <i>x</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce585154e4780be88423541d65e57da942e543e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.682ex; height:5.176ex;" alt="{\displaystyle f(x)={\frac {1}{x}}}"></span></dd></dl> <p>Esta función é unha <a href="/wiki/Hip%C3%A9rbole_(xeometr%C3%ADa)" title="Hipérbole (xeometría)">hipérbole</a> composta por dous tramos. <i>x</i> < 0 e <b>x</b> > 0. Como se ve, efectivamente é continua en todo o <a href="/wiki/Dominio_de_definici%C3%B3n" title="Dominio de definición">dominio</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 \left(-\infty ,0\right)\cup \left(0,+\infty \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>∪</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-\infty ,0\right)\cup \left(0,+\infty \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ecfbd75a8df3c49538c27978fc34d03dea5a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.858ex; height:2.843ex;" alt="{\displaystyle \left(-\infty ,0\right)\cup \left(0,+\infty \right)}"></span> porque non está definida en <i>x</i>= 0. Se se estende o dominio da función a ℝ (dándolle un valor arbitrario a <i>f</i>(0)) a función será descontinua.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span>[</span>8<span>]</span></a></sup>) </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Teoremas_sobre_funcións_continuas"><span id="Teoremas_sobre_funci.C3.B3ns_continuas"></span>Teoremas sobre funcións continuas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=10" title="Editar a sección: «Teoremas sobre funcións continuas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=10" title="Editar o código fonte da sección: Teoremas sobre funcións continuas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Estes son algúns dos teoremas máis importantes sobre funcións continuas. </p> <ol><li><a href="/wiki/Teorema_de_Weierstrass" title="Teorema de Weierstrass">Teorema de Weierstrass</a>: Se <i>f</i> é continua en <span class="mwe-math-element"><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> entón presenta máximos e mínimos absolutos.</li> <li><a href="/w/index.php?title=Teorema_de_Bolzano&action=edit&redlink=1" class="new" title="Teorema de Bolzano (a páxina aínda non existe)">Teorema de Bolzano</a>: Se <i>f</i> é continua en <span class="mwe-math-element"><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> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)f(b)<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a)f(b)<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf828d0f70a94d206b1a11efe2d032da7610fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.664ex; height:2.843ex;" alt="{\displaystyle f(a)f(b)<0}"></span>, entón <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists c\in (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>c</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists c\in (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb37e4dc4d77a073970371ea322d747e9c16758a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.211ex; height:2.843ex;" alt="{\displaystyle \exists c\in (a,b)}"></span> tal que <span class="mwe-math-element"><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(c)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(c)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a6d8bc90109165bcee32d7df3db90b3a3a79c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.356ex; height:2.843ex;" alt="{\displaystyle f(c)=0}"></span>.</li> <li><a href="/wiki/Teorema_do_valor_intermedio" title="Teorema do valor intermedio">Teorema do valor intermedio</a>: Se <i>f</i> é continua en <span class="mwe-math-element"><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> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)<k<f(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a)<k<f(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57b23b1e562ea593b2af035054cb61af6779a0b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.811ex; height:2.843ex;" alt="{\displaystyle f(a)<k<f(b)}"></span> entón <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists c\in (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>c</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists c\in (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb37e4dc4d77a073970371ea322d747e9c16758a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.211ex; height:2.843ex;" alt="{\displaystyle \exists c\in (a,b)}"></span> tal que <span class="mwe-math-element"><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(c)=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(c)=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13a1f588e61b7f0b22f21f86999e9d6a0db7c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.404ex; height:2.843ex;" alt="{\displaystyle f(c)=k}"></span>.</li></ol> <p>Anotación: Se <i>f</i> é unha función sobre un <a href="/wiki/Espazo_compacto" title="Espazo compacto">conxunto compacto</a> entón, a función ten un máximo ou un mínimo. Sobre un conxunto aberto tense o seguinte contraexemplo: a función <span class="mwe-math-element"><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)=1/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> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=1/x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049b4bda6d9e222e496f2670248d9ecfb75841d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.171ex; height:2.843ex;" alt="{\displaystyle f(x)=1/x}"></span> é continua sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span> pero non é limitada. </p> <div class="mw-heading mw-heading3"><h3 id="Derivabilidade_e_continuidade">Derivabilidade e continuidade</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=11" title="Editar a sección: «Derivabilidade e continuidade»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=11" title="Editar o código fonte da sección: Derivabilidade e continuidade"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As funcións derivables son continuas. Se unha <a href="/wiki/Funci%C3%B3n" title="Función">función</a> é <a href="/wiki/Derivada" title="Derivada">derivable</a> en <i>x</i>=<i>a</i> entón é continua en <i>x</i>=<i>a</i>. De modo que a continuidade é unha condición necesaria para a derivabilidade. É dicir, o conxunto das funcións derivables é parte das funcións continuas. </p><p>Cómpre notar que o recíproco non é válido; é dicir que nada se pode afirmar sobre a <a href="/wiki/Derivada" title="Derivada">derivabilidade</a> dunha función continua. Un exemplo claro desta situación é a función <a href="/wiki/Valor_absoluto" title="Valor absoluto">valor absoluto</a> <i>f</i>(<i>x</i>)= |<i>x</i>| que aínda que é continua en todo o seu dominio non é derivable en <i>x</i>=0. Mesmo hai funcións continuas en todo ℝ pero non derivables en ningún punto (as funcións do <a href="/wiki/Movemento_browniano" title="Movemento browniano">movemento browniano</a> verifican isto con probabilidade 1). </p> <div class="mw-heading mw-heading4"><h4 id="Clase_de_continuidade">Clase de continuidade</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=12" title="Editar a sección: «Clase de continuidade»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=12" title="Editar o código fonte da sección: Clase de continuidade"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unha función <span class="mwe-math-element"><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:\Omega \subset \mathbb {R} \longrightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">⟶</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\Omega \subset \mathbb {R} \longrightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6174e8be15c63bd7120dd9e632ecab979ea35f05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.444ex; height:2.509ex;" alt="{\displaystyle f:\Omega \subset \mathbb {R} \longrightarrow \mathbb {R} }"></span>, dise que: </p> <ul><li>é de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{0}(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fd98ca89e4aefa5db5ab936bd873a974511272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.727ex; height:3.176ex;" alt="{\displaystyle C^{0}(\Omega )\,}"></span> cando é continua en todo o dominio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" 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 \Omega }"></span></li> <li>é de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{k}(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f99fafe6ea43e670fdb962ed52f938a2ceeefef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.761ex; height:3.176ex;" alt="{\displaystyle C^{k}(\Omega )\,}"></span> se está definida en todo o dominio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" 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 \Omega }"></span> xunto coas súas derivadas até orde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30d7dcf305b7bce39d36df72fe3985b47aa9961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}"></span> e todas elas son continuas.</li> <li>é de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b9e54228c03b9d8cf87a29000b4f48b89e36d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.548ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\Omega )\,}"></span> se ten derivadas continuas de calquera orde. As funcións deste tipo non son nercesariamente <a href="/w/index.php?title=Funci%C3%B3n_anal%C3%ADtica&action=edit&redlink=1" class="new" title="Función analítica (a páxina aínda non existe)">analíticas</a>.</li> <li>é de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{-1}(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{-1}(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18257fb58d808048574d572a3868ef5f7a2e6524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.005ex; height:3.176ex;" alt="{\displaystyle C^{-1}(\Omega )\,}"></span> se é a derivada no <a href="/w/index.php?title=Teor%C3%ADa_de_distribuci%C3%B3ns&action=edit&redlink=1" class="new" title="Teoría de distribucións (a páxina aínda non existe)">sentido das distribucións</a> dunha función de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{0}(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fd98ca89e4aefa5db5ab936bd873a974511272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.727ex; height:3.176ex;" alt="{\displaystyle C^{0}(\Omega )\,}"></span>.</li> <li>Unha <a href="/w/index.php?title=Funci%C3%B3n_xeneralizada&action=edit&redlink=1" class="new" title="Función xeneralizada (a páxina aínda non existe)">función xeneralizada</a> dise de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{-k}(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{-k}(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1cbd5cb14c4c75258b6fce0c3b17d7765a1c730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.04ex; height:3.176ex;" alt="{\displaystyle C^{-k}(\Omega )\,}"></span> se é a derivada k-ésima no sentido das distribucións dunha función de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{0}(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{0}(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29fd98ca89e4aefa5db5ab936bd873a974511272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.727ex; height:3.176ex;" alt="{\displaystyle C^{0}(\Omega )\,}"></span>.</li></ul> <p>Calquera <a href="/w/index.php?title=Funci%C3%B3n_polin%C3%B3mica&action=edit&redlink=1" class="new" title="Función polinómica (a páxina aínda non existe)">función polinómica</a> dunha variable é unha función de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c913ced284ad05bd42c8f004a3a83dee6323cb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} )}"></span>. A función xeneralizada denominada <a href="/w/index.php?title=Delta_de_Dirac&action=edit&redlink=1" class="new" title="Delta de Dirac (a páxina aínda non existe)">delta de Dirac</a> é unha función de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{-2}(\mathbb {R} )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{-2}(\mathbb {R} )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b491e9138893d6d3a54cdeef22a9244455bf4ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.005ex; height:3.176ex;" alt="{\displaystyle C^{-2}(\mathbb {R} )\,}"></span> xa que é a derivada segunda da función rampla que é continua, e a derivada primeira da <a href="/w/index.php?title=Funci%C3%B3n_en_esqueira_de_Heaviside&action=edit&redlink=1" class="new" title="Función en esqueira de Heaviside (a páxina aínda non existe)">función en esqueira de Heaviside</a> que é de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbeead216a363d09a6d0a05e192bdc3e7ed1067f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.131ex; height:2.676ex;" alt="{\displaystyle C^{-1}}"></span> </p><p>Pódense dar exemplos que mostran que hai funcións de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{k}(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f99fafe6ea43e670fdb962ed52f938a2ceeefef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.761ex; height:3.176ex;" alt="{\displaystyle C^{k}(\Omega )\,}"></span> que non son de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k+1}(\Omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{k+1}(\Omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253b82b8eeb853a207c199d209eddce37df6d0a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:3.176ex;" alt="{\displaystyle C^{k+1}(\Omega )\,}"></span>. Os exemplos clásicos son <span class="mwe-math-element"><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_{k}(x)=x^{k}\operatorname {sen}(1/x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>sen</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{k}(x)=x^{k}\operatorname {sen}(1/x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b17b9d81fa7659f3219c764f0a75200d9ee4ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.976ex; height:3.176ex;" alt="{\displaystyle f_{k}(x)=x^{k}\operatorname {sen}(1/x)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Funcións_continuas_en_espazos_topolóxicos"><span id="Funci.C3.B3ns_continuas_en_espazos_topol.C3.B3xicos"></span>Funcións continuas en espazos topolóxicos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=13" title="Editar a sección: «Funcións continuas en espazos topolóxicos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=13" title="Editar o código fonte da sección: Funcións continuas en espazos topolóxicos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sexan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,T_{X})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,T_{X})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1504dabcbaafec8079720ee585f23b6c63ec393c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.813ex; height:2.843ex;" alt="{\displaystyle (X,T_{X})}"></span> e <span class="mwe-math-element"><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,T_{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y,T_{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256e44962aa4e733fc723a02214cb62980f05cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.46ex; height:2.843ex;" alt="{\displaystyle (Y,T_{Y})}"></span> dous <a href="/wiki/Espazo_topol%C3%B3xico" title="Espazo topolóxico">espazos topolóxicos</a>. Unha <a href="/wiki/Funci%C3%B3n" title="Función">aplicación</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\longrightarrow Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">⟶</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\longrightarrow Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0275c3b59055c93ee51ea1ac2d126a2619151bf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.065ex; height:2.509ex;" alt="{\displaystyle f:X\longrightarrow Y}"></span> dise que é continua se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/406c1fafd58adcd4d71fc415677952a867d22e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.289ex; height:3.176ex;" alt="{\displaystyle f^{-1}(G)}"></span> é un aberto de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, calquera que sexa o <a href="/wiki/Conxunto_aberto" title="Conxunto aberto">aberto</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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>. Esta é a continudade vista globalmente, a que segue é a continuidade nun punto do dominio. </p><p>Esta definición redúcese á definición ordinaria de continuidade dunha función <span class="mwe-math-element"><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:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad78382c3d23bcb4051b3148f1a23b1d0ba52e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.079ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}"></span> se sobre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> e <span class="mwe-math-element"><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} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}"></span> se considera a topoloxía inducida pola distancia euclidiana. </p><p>Coa mesma notación anterior, se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>, dise que <span class="mwe-math-element"><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> é continua en <span class="mwe-math-element"><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> cando se obtén que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b5171cc2674480baf68e55ff20378a5413d4728" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.25ex; height:3.176ex;" alt="{\displaystyle f^{-1}(V)}"></span> é unha <a href="/w/index.php?title=Veci%C3%B1anza_(topolox%C3%ADa)&action=edit&redlink=1" class="new" title="Veciñanza (topoloxía) (a páxina aínda non existe)">veciñanza</a> de <span class="mwe-math-element"><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>, calquera que sexa a veciñanza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> de <span class="mwe-math-element"><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>. </p><p>É inmediato entón comprobar que <span class="mwe-math-element"><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> é continua se e só se é continua en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>, calquera que sexa este, é dicir, cando sexa continua en cada un dos puntos do seu dominio. </p> <div class="mw-heading mw-heading2"><h2 id="Funcións_continuas_sobre_os_números_ordinais"><span id="Funci.C3.B3ns_continuas_sobre_os_n.C3.BAmeros_ordinais"></span>Funcións continuas sobre os números ordinais</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=14" title="Editar a sección: «Funcións continuas sobre os números ordinais»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=14" title="Editar o código fonte da sección: Funcións continuas sobre os números ordinais"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O termo función continua na parte da <a href="/wiki/Teor%C3%ADa_de_conxuntos" title="Teoría de conxuntos">teoría de conxuntos</a> que se refire aos <a href="/wiki/N%C3%BAmero_ordinal" title="Número ordinal">números ordinais</a> ten un sentido diferente ao referido ás funcións sobre espazos topolóxicos. Concretamente unha función <i>F</i> definida sobre a <a href="/w/index.php?title=Clase_(teor%C3%ADa_de_conxuntos)&action=edit&redlink=1" class="new" title="Clase (teoría de conxuntos) (a páxina aínda non existe)">clase</a> dos números ordinais <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {On} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {On} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3720993ab82325279db5cae4b5c33066ed839b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.101ex; height:2.176ex;" alt="{\displaystyle \mathrm {On} }"></span> é continua se para cada ordinal límite γ se cumpre a seguinte propiedade: </p> <blockquote style="padding: 5px 10px;background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em"> <p><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(\gamma )=\bigcup \{F(\sigma )|\ \sigma <\gamma ,\ \sigma \in \mathrm {On} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>⋃</mo> <mo fence="false" stretchy="false">{</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>σ</mi> <mo><</mo> <mi>γ</mi> <mo>,</mo> <mtext> </mtext> <mi>σ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">n</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\gamma )=\bigcup \{F(\sigma )|\ \sigma <\gamma ,\ \sigma \in \mathrm {On} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef285736653e23dab0952b8a3f9145466ea3a56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:33.501ex; height:3.843ex;" alt="{\displaystyle F(\gamma )=\bigcup \{F(\sigma )|\ \sigma <\gamma ,\ \sigma \in \mathrm {On} \}}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Notas">Notas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=15" title="Editar a sección: «Notas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=15" title="Editar o código fonte da sección: Notas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation book">Bolzano, Bernard (1817). <i>Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewaehren, wenigstens eine reele Wurzel der Gleichung liege</i>. Prague: Haase.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AFunci%C3%B3n+continua&rft.aufirst=Bernard&rft.aulast=Bolzano&rft.btitle=Rein+analytischer+Beweis+des+Lehrsatzes+dass+zwischen+je+zwey+Werthen%2C+die+ein+entgegengesetztes+Resultat+gewaehren%2C+wenigstens+eine+reele+Wurzel+der+Gleichung+liege&rft.date=1817&rft.genre=book&rft.place=Prague&rft.pub=Haase&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book">Dugac, Pierre (1973). <i>Eléments d'Analyse de Karl Weierstrass</i>. <i>Archive for History of Exact Sciences</i> <b>10</b>. pp. 41–176. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2Fbf00343406">10.1007/bf00343406</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AFunci%C3%B3n+continua&rft.aufirst=Pierre&rft.aulast=Dugac&rft.btitle=El%C3%A9ments+d%27Analyse+de+Karl+Weierstrass&rft.date=1973&rft.genre=book&rft.pages=41-176&rft_id=info%3Adoi%2F10.1007%2Fbf00343406&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite class="citation book">Goursat, E. (1904). <a rel="nofollow" class="external text" href="https://archive.org/details/acourseinmathem00unkngoog"><i>A course in mathematical analysis</i></a>. Boston: Ginn. p. <a rel="nofollow" class="external text" href="https://archive.org/details/acourseinmathem00unkngoog/page/n18">2</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AFunci%C3%B3n+continua&rft.aufirst=E.&rft.aulast=Goursat&rft.btitle=A+course+in+mathematical+analysis&rft.date=1904&rft.genre=book&rft.pages=2&rft.place=Boston&rft.pub=Ginn&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Facourseinmathem00unkngoog&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation book">Jordan, M.C. (1893). <i>Cours d'analyse de l'École polytechnique</i> <b>1</b> (2nd ed.). Paris: Gauthier-Villars. p. 46.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AFunci%C3%B3n+continua&rft.aufirst=M.C.&rft.aulast=Jordan&rft.btitle=Cours+d%27analyse+de+l%27%C3%89cole+polytechnique&rft.date=1893&rft.edition=2nd&rft.genre=book&rft.pages=46&rft.place=Paris&rft.pub=Gauthier-Villars&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><cite class="citation book">Harper, J.F. (2016). <i>Defining continuity of real functions of real variables</i>. <i>BSHM Bulletin: Journal of the British Society for the History of Mathematics</i>. pp. 1–16. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F17498430.2015.1116053">10.1080/17498430.2015.1116053</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AFunci%C3%B3n+continua&rft.aufirst=J.F.&rft.aulast=Harper&rft.btitle=Defining+continuity+of+real+functions+of+real+variables&rft.date=2016&rft.genre=book&rft.pages=1-16&rft_id=info%3Adoi%2F10.1080%2F17498430.2015.1116053&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><cite class="citation book">Rusnock, P.; Kerr-Lawson, A. (2005). <i>Bolzano and uniform continuity</i>. <i>Historia Mathematica</i> <b>32</b>. pp. 303–311. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2Fj.hm.2004.11.003">10.1016/j.hm.2004.11.003</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AFunci%C3%B3n+continua&rft.au=Kerr-Lawson%2C+A.&rft.aufirst=P.&rft.aulast=Rusnock&rft.btitle=Bolzano+and+uniform+continuity&rft.date=2005&rft.genre=book&rft.pages=303-311&rft_id=info%3Adoi%2F10.1016%2Fj.hm.2004.11.003&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><cite class="citation book"><a href="/w/index.php?title=Serge_Lang&action=edit&redlink=1" class="new" title="Serge Lang (a páxina aínda non existe)">Lang, Serge</a> (1997). <i>Undergraduate analysis</i>. <a href="/w/index.php?title=Undergraduate_Texts_in_Mathematics&action=edit&redlink=1" class="new" title="Undergraduate Texts in Mathematics (a páxina aínda non existe)">Undergraduate Texts in Mathematics</a> (2nd ed.). Berlín, Nova York: <a href="/w/index.php?title=Springer-Verlag&action=edit&redlink=1" class="new" title="Springer-Verlag (a páxina aínda non existe)">Springer-Verlag</a>. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:Fontes_bibliogr%C3%A1ficas/978-0-387-94841-6" title="Especial:Fontes bibliográficas/978-0-387-94841-6">978-0-387-94841-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AFunci%C3%B3n+continua&rft.aufirst=Serge&rft.aulast=Lang&rft.btitle=Undergraduate+analysis&rft.date=1997&rft.edition=2nd&rft.genre=book&rft.isbn=978-0-387-94841-6&rft.place=Berl%C3n%2C+Nova+York&rft.pub=Springer-Verlag&rft.series=Undergraduate+Texts+in+Mathematics&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>, section II.4</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><cite class="citation web">Speck, Jared (2014). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161006014646/http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf">"Continuity and Discontinuity"</a> <span style="font-size:85%;">(PDF)</span>. <i>MIT Math</i>. p. 3. Arquivado dende <a rel="nofollow" class="external text" href="http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf">o orixinal</a> <span style="font-size:85%;">(PDF)</span> o 06-10-2016<span class="reference-accessdate">. Consultado o 2-9-2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AFunci%C3%B3n+continua&rft.atitle=Continuity+and+Discontinuity&rft.aufirst=Jared&rft.aulast=Speck&rft.date=2014&rft.genre=unknown&rft.jtitle=MIT+Math&rft.pages=3&rft_id=http%3A%2F%2Fmath.mit.edu%2F~jspeck%2F18.01_Fall%25202014%2FSupplementary%2520notes%2F01c.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Véxase_tamén"><span id="V.C3.A9xase_tam.C3.A9n"></span>Véxase tamén</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=16" title="Editar a sección: «Véxase tamén»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=16" title="Editar o código fonte da sección: Véxase tamén"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table role="presentation" class="mbox-small plainlinks sistersitebox" style="background-color:var(--background-color-neutral-subtle, #f8f9fa);border:1px solid var(--border-color-base, #a2a9b1);color:inherit"> <tbody><tr> <td class="mbox-image"><span class="noviewer" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></td> <td class="mbox-text plainlist"><a href="https://commons.wikimedia.org/wiki/Portada_galega" class="extiw" title="commons:Portada galega">Wikimedia Commons</a> ten máis contidos multimedia na categoría:  <i><b><a href="https://commons.wikimedia.org/wiki/Special:Search/Funci%C3%B3n_continua" class="extiw" title="commons:Special:Search/Función continua">Función continua</a></b></i></td></tr> </tbody></table> <div class="mw-heading mw-heading3"><h3 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=17" title="Editar a sección: «Bibliografía»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=17" title="Editar o código fonte da sección: Bibliografía"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Serge Lang (1990): <i>Introdución al análisis Matemático</i> , Wilmington Delaware.</li> <li>James R. Munkres (2002): <i>Topología</i>, Madrid.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Outros_artigos">Outros artigos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=18" title="Editar a sección: «Outros artigos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=18" title="Editar o código fonte da sección: Outros artigos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Clasificaci%C3%B3n_de_descontinuidades&action=edit&redlink=1" class="new" title="Clasificación de descontinuidades (a páxina aínda non existe)">Clasificación de descontinuidades</a></li> <li><a href="/wiki/Lista_de_funci%C3%B3ns_matem%C3%A1ticas" title="Lista de funcións matemáticas">Lista de funcións matemáticas</a></li> <li><a href="/wiki/Derivaci%C3%B3n_(matem%C3%A1tica)" title="Derivación (matemática)">Derivación</a></li> <li><a href="/wiki/Continuo" title="Continuo">Continuo</a></li> <li><a href="/wiki/Continuidade_uniforme" title="Continuidade uniforme">Continuidade uniforme</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ligazóns_externas"><span id="Ligaz.C3.B3ns_externas"></span>Ligazóns externas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Funci%C3%B3n_continua&veaction=edit&section=19" title="Editar a sección: «Ligazóns externas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Funci%C3%B3n_continua&action=edit&section=19" title="Editar o código fonte da sección: Ligazóns externas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://archives.math.utk.edu/visual.calculus/">Visual Calculus</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110924071226/http://archives.math.utk.edu/visual.calculus/">Arquivado</a> 24 de setembro de 2011 en <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. by Lawrence S. Husch, <a href="/w/index.php?title=University_of_Tennessee&action=edit&redlink=1" class="new" title="University of Tennessee (a páxina aínda non existe)">University of Tennessee</a> (2001).</li></ul> <div role="navigation" class="navbox" aria-labelledby="Control_de_autoridades" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_de_autoridades" scope="row" class="navbox-group" style="width:1%;width: 12%; text-align:center;"><a href="/wiki/Axuda:Control_de_autoridades" title="Axuda:Control de autoridades">Control de autoridades</a></th><td class="navbox-list navbox-odd plainlinks" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Wikidata:Main_Page" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span>: <span class="uid"><a href="https://www.wikidata.org/wiki/Q170058" class="extiw" title="wikidata:Q170058">Q170058</a></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_Central_de_Florencia" title="Biblioteca Nacional Central de Florencia">BNCF</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=53874">53874</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_de_Francia" title="Biblioteca Nacional de Francia">BNF</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb12123565q">12123565q</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Grande_Enciclopedia_Rusa" title="Grande Enciclopedia Rusa">BRE</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://old.bigenc.ru/text/2261813">2261813</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">EBID</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/continuous-function">ID</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Gran_Enciclop%C3%A8dia_Catalana" title="Gran Enciclopèdia Catalana">GEC</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://www.enciclopedia.cat/ec-gec-0153274.xml">0153274</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4183162-7">4183162-7</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85052334">sh85052334</a></span></span></span></li> <li><span style="white-space:nowrap;">MAG: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://web.archive.org/web/*/https://academic.microsoft.com/v2/detail/184825909">184825909</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_t%C3%A9cnica_nacional_de_Praga" title="Biblioteca técnica nacional de Praga">PSH</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://psh.techlib.cz/skos/PSH7382">7382</a></span></span></span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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