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class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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="Toggle the table of contents" > <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 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Available in 41 languages" > <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-41" 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">41 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <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%B7%D8%A7%D8%A8%D9%82%D8%A9" title="دالة مطابقة – Arabic" lang="ar" hreflang="ar" data-title="دالة مطابقة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Funkcija_identiteta" title="Funkcija identiteta – Bosnian" lang="bs" hreflang="bs" data-title="Funkcija identiteta" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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_identitat" title="Funció identitat – Catalan" lang="ca" hreflang="ca" data-title="Funció identitat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C3%87%D0%B0%D0%B2%D0%B0%D1%85%D0%BB%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8" title="Çавахла функци – Chuvash" lang="cv" hreflang="cv" data-title="Çавахла функци" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Identita_(matematika)" title="Identita (matematika) – Czech" lang="cs" hreflang="cs" data-title="Identita (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Funzioni_idantit%C3%A0" title="Funzioni idantità – Corsican" lang="co" hreflang="co" data-title="Funzioni idantità" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target"><span>Corsu</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Identitetsfunktion" title="Identitetsfunktion – Danish" lang="da" hreflang="da" data-title="Identitetsfunktion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Identische_Abbildung" title="Identische Abbildung – German" lang="de" hreflang="de" data-title="Identische Abbildung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Samasusteisendus" title="Samasusteisendus – Estonian" lang="et" hreflang="et" data-title="Samasusteisendus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B1%CF%85%CF%84%CE%BF%CF%84%CE%B9%CE%BA%CE%AE_%CF%83%CF%85%CE%BD%CE%AC%CF%81%CF%84%CE%B7%CF%83%CE%B7" title="Ταυτοτική συνάρτηση – Greek" lang="el" hreflang="el" data-title="Ταυτοτική συνάρτηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_identidad" title="Función identidad – Spanish" lang="es" hreflang="es" data-title="Función identidad" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Identa_funkcio" title="Identa funkcio – Esperanto" lang="eo" hreflang="eo" data-title="Identa funkcio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Identitate_funtzio" title="Identitate funtzio – Basque" lang="eu" hreflang="eu" data-title="Identitate funtzio" data-language-autonym="Euskara" data-language-local-name="Basque" 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%87%D9%85%D8%A7%D9%86%DB%8C" title="تابع همانی – Persian" lang="fa" hreflang="fa" data-title="تابع همانی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Application_identit%C3%A9" title="Application identité – French" lang="fr" hreflang="fr" data-title="Application identité" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Funci%C3%B3n_identidade" title="Función identidade – Galician" lang="gl" hreflang="gl" data-title="Función identidade" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%AD%EB%93%B1_%ED%95%A8%EC%88%98" title="항등 함수 – Korean" lang="ko" hreflang="ko" data-title="항등 함수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A4%A4%E0%A5%8D%E0%A4%B8%E0%A4%AE%E0%A4%95_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="तत्समक फलन – Hindi" lang="hi" hreflang="hi" data-title="तत्समक 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/Funkcija_identiteta" title="Funkcija identiteta – Croatian" lang="hr" hreflang="hr" data-title="Funkcija identiteta" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fungsi_identitas" title="Fungsi identitas – Indonesian" lang="id" hreflang="id" data-title="Fungsi identitas" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Samsemdarv%C3%B6rpun" title="Samsemdarvörpun – Icelandic" lang="is" hreflang="is" data-title="Samsemdarvörpun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_identit%C3%A0" title="Funzione identità – Italian" lang="it" hreflang="it" data-title="Funzione identità" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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%99%D7%AA_%D7%94%D7%96%D7%94%D7%95%D7%AA" title="פונקציית הזהות – Hebrew" lang="he" hreflang="he" data-title="פונקציית הזהות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Aplicaziun_identica" title="Aplicaziun identica – Lombard" lang="lmo" hreflang="lmo" data-title="Aplicaziun identica" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Identieke_afbeelding" title="Identieke afbeelding – Dutch" lang="nl" hreflang="nl" data-title="Identieke afbeelding" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%8F%E0%A4%95%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%AE%E0%A4%95_%E0%A4%AB%E0%A4%B2%E0%A4%A8" title="एकात्मक फलन – Nepali" lang="ne" hreflang="ne" data-title="एकात्मक फलन" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%81%92%E7%AD%89%E5%86%99%E5%83%8F" title="恒等写像 – Japanese" lang="ja" hreflang="ja" data-title="恒等写像" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Funkcja_to%C5%BCsamo%C5%9Bciowa" title="Funkcja tożsamościowa – Polish" lang="pl" hreflang="pl" data-title="Funkcja tożsamościowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_identidade" title="Função identidade – Portuguese" lang="pt" hreflang="pt" data-title="Função identidade" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Identitate_(matematic%C4%83)" title="Identitate (matematică) – Romanian" lang="ro" hreflang="ro" data-title="Identitate (matematică)" data-language-autonym="Română" data-language-local-name="Romanian" 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%A2%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%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="Тождественное отображение – Russian" lang="ru" hreflang="ru" data-title="Тождественное отображение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Identi%C4%8Dna_funkcija" title="Identična funkcija – Slovenian" lang="sl" hreflang="sl" data-title="Identična funkcija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Identiteettifunktio" title="Identiteettifunktio – Finnish" lang="fi" hreflang="fi" data-title="Identiteettifunktio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Identitetsfunktion" title="Identitetsfunktion – Swedish" lang="sv" hreflang="sv" data-title="Identitetsfunktion" data-language-autonym="Svenska" data-language-local-name="Swedish" 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%AE%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%8A%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AF%88%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81" title="முற்றொருமைச் சார்பு – Tamil" lang="ta" hreflang="ta" data-title="முற்றொருமைச் சார்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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%B9%80%E0%B8%AD%E0%B8%81%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B9%8C" title="ฟังก์ชันเอกลักษณ์ – Thai" lang="th" hreflang="th" data-title="ฟังก์ชันเอกลักษณ์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Birim_fonksiyon" title="Birim fonksiyon – Turkish" lang="tr" hreflang="tr" data-title="Birim fonksiyon" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%BE%D1%82%D0%BE%D0%B6%D0%BD%D0%B5_%D0%B2%D1%96%D0%B4%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F" title="Тотожне відображення – Ukrainian" lang="uk" hreflang="uk" data-title="Тотожне відображення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%A0m_%C4%91%E1%BB%93ng_nh%E1%BA%A5t" title="Hàm đồng nhất – Vietnamese" lang="vi" hreflang="vi" data-title="Hàm đồng nhất" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%81%86%E7%AD%89%E5%87%BD%E6%95%B8" title="恆等函數 – Cantonese" lang="yue" hreflang="yue" data-title="恆等函數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%81%86%E7%AD%89%E5%87%BD%E6%95%B8" title="恆等函數 – Chinese" lang="zh" hreflang="zh" data-title="恆等函數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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/Q321119#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> 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.hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Null_function" title="Null function">Null function</a> or <a href="/wiki/Empty_function" class="mw-redirect" title="Empty function">Empty function</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Function-x.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Function-x.svg/220px-Function-x.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Function-x.svg/330px-Function-x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Function-x.svg/440px-Function-x.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph</a> of the identity function on the <a href="/wiki/Real_number" title="Real number">real numbers</a></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>identity function</b>, also called an <b>identity relation</b>, <b>identity map</b> or <b>identity transformation</b>, is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that always returns the value that was used as its <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a>, unchanged. That is, when <span class="texhtml mvar" style="font-style:italic;">f</span> is the identity function, the <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a> <span class="texhtml"><i>f</i>(<i>x</i>) = <i>x</i></span> is true for all values of <span class="texhtml mvar" style="font-style:italic;">x</span> to which <span class="texhtml mvar" style="font-style:italic;">f</span> can be applied. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_function&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formally, if <span class="texhtml"><i>X</i></span> is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, the identity function <span class="texhtml"><i>f</i></span> on <span class="texhtml"><i>X</i></span> is defined to be a function with <span class="texhtml"><i>X</i></span> as its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> and <a href="/wiki/Codomain" title="Codomain">codomain</a>, satisfying </p> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.6em;"><span class="texhtml"><i>f</i>(<i>x</i>) = <i>x</i></span>   for all elements <span class="texhtml"><i>x</i></span> in <span class="texhtml"><i>X</i></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></div> <p>In other words, the function value <span class="texhtml"><i>f</i>(<i>x</i>)</span> in the codomain <span class="texhtml"><i>X</i></span> is always the same as the input element <span class="texhtml"><i>x</i></span> in the domain <span class="texhtml"><i>X</i></span>. The identity function on <span class="texhtml mvar" style="font-style:italic;">X</span> is clearly an <a href="/wiki/Injective_function" title="Injective function">injective function</a> as well as a <a href="/wiki/Surjective_function" title="Surjective function">surjective function</a> (its codomain is also its <a href="/wiki/Range_(function)" class="mw-redirect" title="Range (function)">range</a>), so it is <a href="/wiki/Bijection" title="Bijection">bijective</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The identity function <span class="texhtml"><i>f</i></span> on <span class="texhtml"><i>X</i></span> is often denoted by <span class="texhtml">id<sub><i>X</i></sub></span>. </p><p>In <a href="/wiki/Set_theory" title="Set theory">set theory</a>, where a function is defined as a particular kind of <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a>, the identity function is given by the <a href="/wiki/Identity_relation" class="mw-redirect" title="Identity relation">identity relation</a>, or <i>diagonal</i> of <span class="texhtml"><i>X</i></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Algebraic_properties">Algebraic properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_function&action=edit&section=2" title="Edit section: Algebraic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml"><i>f</i> : <i>X</i> → <i>Y</i></span> is any function, then <span class="texhtml"><i>f</i> ∘ id<sub><i>X</i></sub> = <i>f</i> = id<sub><i>Y</i></sub> ∘ <i>f</i></span>, where "∘" denotes <a href="/wiki/Function_composition" title="Function composition">function composition</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In particular, <span class="texhtml">id<sub><i>X</i></sub></span> is the <a href="/wiki/Identity_element" title="Identity element">identity element</a> of the <a href="/wiki/Monoid" title="Monoid">monoid</a> of all functions from <span class="texhtml"><i>X</i></span> to <span class="texhtml"><i>X</i></span> (under function composition). </p><p>Since the identity element of a monoid is <a href="/wiki/Unique_(mathematics)" class="mw-redirect" title="Unique (mathematics)">unique</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> one can alternately define the identity function on <span class="texhtml"><i>M</i></span> to be this identity element. Such a definition generalizes to the concept of an <a href="/wiki/Identity_morphism" class="mw-redirect" title="Identity morphism">identity morphism</a> in <a href="/wiki/Category_theory" title="Category theory">category theory</a>, where the <a href="/wiki/Endomorphism" title="Endomorphism">endomorphisms</a> of <span class="texhtml"><i>M</i></span> need not be functions. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_function&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The identity function is a <a href="/wiki/Linear_map" title="Linear map">linear operator</a> when applied to <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>In an <span class="texhtml mvar" style="font-style:italic;">n</span>-<a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimensional</a> <a href="/wiki/Vector_space" title="Vector space">vector space</a> the identity function is represented by the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> <span class="texhtml"><i>I</i><sub><i>n</i></sub></span>, regardless of the <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> chosen for the space.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>The identity function on the positive <a href="/wiki/Integer" title="Integer">integers</a> is a <a href="/wiki/Completely_multiplicative_function" title="Completely multiplicative function">completely multiplicative function</a> (essentially multiplication by 1), considered in <a href="/wiki/Number_theory" title="Number theory">number theory</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li>In a <a href="/wiki/Metric_space" title="Metric space">metric space</a> the identity function is trivially an <a href="/wiki/Isometry" title="Isometry">isometry</a>. An object without any <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> has as its <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> the <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a> containing only this isometry (symmetry type <span class="texhtml">C<sub>1</sub></span>).<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>In a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, the identity function is always <a href="/wiki/Continuous_function#Continuous_functions_between_topological_spaces" title="Continuous function">continuous</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>The identity function is <a href="/wiki/Idempotence" title="Idempotence">idempotent</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_function&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Identity_matrix" title="Identity matrix">Identity matrix</a></li> <li><a href="/wiki/Inclusion_map" title="Inclusion map">Inclusion map</a></li> <li><a href="/wiki/Indicator_function" title="Indicator function">Indicator function</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Identity_function&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFKnapp2006" class="citation cs2">Knapp, Anthony W. (2006), <i>Basic algebra</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-3248-9" title="Special:BookSources/978-0-8176-3248-9"><bdi>978-0-8176-3248-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra&rft.pub=Springer&rft.date=2006&rft.isbn=978-0-8176-3248-9&rft.aulast=Knapp&rft.aufirst=Anthony+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMapa2014" class="citation book cs1">Mapa, Sadhan Kumar (7 April 2014). <i>Higher Algebra Abstract and Linear</i> (11th ed.). Sarat Book House. p. 36. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-93-80663-24-1" title="Special:BookSources/978-93-80663-24-1"><bdi>978-93-80663-24-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Higher+Algebra+Abstract+and+Linear&rft.pages=36&rft.edition=11th&rft.pub=Sarat+Book+House&rft.date=2014-04-07&rft.isbn=978-93-80663-24-1&rft.aulast=Mapa&rft.aufirst=Sadhan+Kumar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=oIFLAQAAIAAJ&q=the+identity+function+is+given+by+the+identity+relation,+or+diagonal"><i>Proceedings of Symposia in Pure Mathematics</i></a>. American Mathematical Society. 1974. p. 92. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1425-3" title="Special:BookSources/978-0-8218-1425-3"><bdi>978-0-8218-1425-3</bdi></a>. <q>...then the diagonal set determined by M is the identity relation...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proceedings+of+Symposia+in+Pure+Mathematics&rft.pages=92&rft.pub=American+Mathematical+Society&rft.date=1974&rft.isbn=978-0-8218-1425-3&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoIFLAQAAIAAJ%26q%3Dthe%2Bidentity%2Bfunction%2Bis%2Bgiven%2Bby%2Bthe%2Bidentity%2Brelation%2C%2Bor%2Bdiagonal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNel2016" class="citation book cs1">Nel, Louis (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_JdPDAAAQBAJ&pg=PA21"><i>Continuity Theory</i></a>. p. 21. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-31159-3">10.1007/978-3-319-31159-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-31159-3" title="Special:BookSources/978-3-319-31159-3"><bdi>978-3-319-31159-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Continuity+Theory&rft.pages=21&rft.date=2016&rft_id=info%3Adoi%2F10.1007%2F978-3-319-31159-3&rft.isbn=978-3-319-31159-3&rft.aulast=Nel&rft.aufirst=Louis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_JdPDAAAQBAJ%26pg%3DPA21&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosalesGarcía-Sánchez1999" class="citation book cs1">Rosales, J. C.; García-Sánchez, P. A. (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LQsH6m-x8ysC&q=identity+element+of+a+monoid+is+unique&pg=PA1"><i>Finitely Generated Commutative Monoids</i></a>. Nova Publishers. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56072-670-8" title="Special:BookSources/978-1-56072-670-8"><bdi>978-1-56072-670-8</bdi></a>. <q>The element 0 is usually referred to as the identity element and if it exists, it is unique</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finitely+Generated+Commutative+Monoids&rft.pages=1&rft.pub=Nova+Publishers&rft.date=1999&rft.isbn=978-1-56072-670-8&rft.aulast=Rosales&rft.aufirst=J.+C.&rft.au=Garc%C3%ADa-S%C3%A1nchez%2C+P.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLQsH6m-x8ysC%26q%3Didentity%2Belement%2Bof%2Ba%2Bmonoid%2Bis%2Bunique%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton2005" class="citation cs2">Anton, Howard (2005), <i>Elementary Linear Algebra (Applications Version)</i> (9th ed.), Wiley International</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra+%28Applications+Version%29&rft.edition=9th&rft.pub=Wiley+International&rft.date=2005&rft.aulast=Anton&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFT._S._Shores2007" class="citation book cs1">T. S. Shores (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8qwTb9P-iW8C&q=Matrix+Analysis"><i>Applied Linear Algebra and Matrix Analysis</i></a>. Undergraduate Texts in Mathematics. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-038-733-195-9" title="Special:BookSources/978-038-733-195-9"><bdi>978-038-733-195-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Linear+Algebra+and+Matrix+Analysis&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2007&rft.isbn=978-038-733-195-9&rft.au=T.+S.+Shores&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8qwTb9P-iW8C%26q%3DMatrix%2BAnalysis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD._MarshallE._OdellM._Starbird2007" class="citation book cs1">D. Marshall; E. Odell; M. Starbird (2007). <i>Number Theory through Inquiry</i>. Mathematical Association of America Textbooks. Mathematical Assn of Amer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0883857519" title="Special:BookSources/978-0883857519"><bdi>978-0883857519</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+through+Inquiry&rft.series=Mathematical+Association+of+America+Textbooks&rft.pub=Mathematical+Assn+of+Amer&rft.date=2007&rft.isbn=978-0883857519&rft.au=D.+Marshall&rft.au=E.+Odell&rft.au=M.+Starbird&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r920966791">.mw-parser-output span.smallcaps{font-variant:small-caps}.mw-parser-output span.smallcaps-smaller{font-size:85%}</style><span class="smallcaps">James W. Anderson</span>, <i>Hyperbolic Geometry</i>, Springer 2005, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-85233-934-9" title="Special:BookSources/1-85233-934-9">1-85233-934-9</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConover2014" class="citation book cs1">Conover, Robert A. (2014-05-21). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KCziAgAAQBAJ&q=identity+function+is+always+continuous&pg=PA65"><i>A First Course in Topology: An Introduction to Mathematical Thinking</i></a>. Courier Corporation. p. 65. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-78001-6" title="Special:BookSources/978-0-486-78001-6"><bdi>978-0-486-78001-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Topology%3A+An+Introduction+to+Mathematical+Thinking&rft.pages=65&rft.pub=Courier+Corporation&rft.date=2014-05-21&rft.isbn=978-0-486-78001-6&rft.aulast=Conover&rft.aufirst=Robert+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKCziAgAAQBAJ%26q%3Didentity%2Bfunction%2Bis%2Balways%2Bcontinuous%26pg%3DPA65&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConferences1968" class="citation book cs1">Conferences, University of Michigan Engineering Summer (1968). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AvAfAAAAMAAJ&q=The+identity+function+is+idempotent."><i>Foundations of Information Systems Engineering</i></a>. <q>we see that an identity element of a semigroup is idempotent.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Information+Systems+Engineering&rft.date=1968&rft.aulast=Conferences&rft.aufirst=University+of+Michigan+Engineering+Summer&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAvAfAAAAMAAJ%26q%3DThe%2Bidentity%2Bfunction%2Bis%2Bidempotent.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdentity+function" class="Z3988"></span></span> </li> </ol></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output 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title="History of the function concept">History</a></li> <li><a href="/wiki/List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean-valued_function" title="Boolean-valued function"><span class="texhtml">X → 𝔹</span></a></li> <li><a href="/wiki/Ordered_pair" title="Ordered pair"><span class="texhtml">𝔹 → X</span></a></li> <li><a href="/wiki/Boolean_function" title="Boolean function"><span class="texhtml">𝔹ⁿ → X</span></a></li> <li><a href="/wiki/Integer-valued_function" title="Integer-valued function"><span class="texhtml">X → ℤ</span></a></li> <li><a href="/wiki/Sequence" title="Sequence"><span class="texhtml">ℤ → X</span></a></li> <li><a href="/wiki/Real-valued_function" title="Real-valued function"><span class="texhtml">X → ℝ</span></a></li> <li><a href="/wiki/Function_of_a_real_variable" title="Function of a real variable"><span class="texhtml">ℝ → X</span></a></li> <li><a href="/wiki/Function_of_several_real_variables" title="Function of several real variables"><span class="texhtml">ℝⁿ → X</span></a></li> <li><a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function"><span class="texhtml">X → ℂ</span></a></li> <li><a href="/wiki/Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable"><span class="texhtml">ℂ → X</span></a></li> <li><a href="/wiki/Function_of_several_complex_variables" title="Function of several complex variables"><span class="texhtml">ℂⁿ → X</span></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Constant_function" title="Constant function">Constant</a></li> <li><a class="mw-selflink selflink">Identity</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear</a></li> <li><a href="/wiki/Polynomial" title="Polynomial">Polynomial</a></li> <li><a href="/wiki/Rational_function" title="Rational function">Rational</a></li> <li><a href="/wiki/Algebraic_function" title="Algebraic function">Algebraic</a></li> <li><a href="/wiki/Analytic_function" title="Analytic function">Analytic</a></li> <li><a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable</a></li> <li><a href="/wiki/Injective_function" title="Injective function">Injective</a></li> <li><a href="/wiki/Surjective_function" title="Surjective function">Surjective</a></li> <li><a href="/wiki/Bijection" title="Bijection">Bijective</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li> <li><a href="/wiki/Function_composition" title="Function composition">Composition</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">λ</a></li> <li><a href="/wiki/Inverse_function" title="Inverse function">Inverse</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a>)</li> <li><a href="/wiki/Set-valued_function" title="Set-valued function">Set-valued</a></li> <li><a href="/wiki/Multivalued_function" title="Multivalued function">Multivalued</a></li> <li><a href="/wiki/Partial_function" title="Partial function">Partial</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit</a></li> <li><a href="/wiki/Function_space" title="Function space">Space</a></li> <li><a href="/wiki/Higher-order_function" title="Higher-order function">Higher-order</a></li> <li><a href="/wiki/Morphism" title="Morphism">Morphism</a></li> <li><a href="/wiki/Functor" title="Functor">Functor</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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