GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Erweiterte Funktion</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Erweiterte Funktion</span></h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Eine <b>erweiterte Funktion</b> und der damit eng verbundene Begriff einer <b>echten Funktion</b> ist eine <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a>, deren <a href="Bild_(Mathematik)" title="Bild (Mathematik)">Wertebereich</a> um den symbolischen Wert <a href="Unendlich" class="mw-redirect" title="Unendlich">unendlich</a> erweitert wird. Dies erleichtert den Umgang mit der Funktion, da man sich auf die <a href="Urbildmenge" class="mw-redirect" title="Urbildmenge">Urbildmengen</a> von Interesse konzentrieren kann, allen anderen Mengen wird der Funktionswert unendlich zugewiesen. Dadurch kann unter Umständen auf Fallunterscheidungen verzichtet werden.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Gegeben ist eine Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon V\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon V\to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38993a46498ca8a11bdc181580a4949ea728cfe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.392ex; height:2.509ex;" alt="{\displaystyle f\colon V\to \mathbb {R} }" loading="lazy"></span> sowie eine Menge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\subset V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\subset V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/096e9772f3d4b1bba3a4e6a5b38aac791ab226ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.328ex; height:2.176ex;" alt="{\displaystyle M\subset V}" loading="lazy"></span>, auf der die Funktion eine gewisse Eigenschaft von Interesse besitzt. Dann heißt die Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}\colon V\to \mathbb {R} \cup \{{+\infty }\}}">

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}\colon V\to \mathbb {R} \cup \{{+\infty }\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/133aca60313606133baac0cc0f05097b95eb76a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.852ex; height:3.176ex;" alt="{\displaystyle {\tilde {f}}\colon V\to \mathbb {R} \cup \{{+\infty }\}}" loading="lazy"></span> mit

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> falls </mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mtext> sonst</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}(x)={\begin{cases}f(x)&{\text{ falls }}x\in M\\{+\infty }&{\text{ sonst}}\end{cases}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21860a55bffbd5744283316ac89d7124a382f355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.029ex; height:6.176ex;" alt="{\displaystyle {\tilde {f}}(x)={\begin{cases}f(x)&{\text{ falls }}x\in M\\{+\infty }&{\text{ sonst}}\end{cases}}}" loading="lazy"></span></dd></dl>

<p><i>erweiterte Funktion</i> zu <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span>. Typische Eigenschaften von Interesse sind zum Beispiel <a href="Reelle_monotone_Funktion" class="mw-redirect" title="Reelle monotone Funktion">Monotonie</a>, <a href="Konvexe_und_konkave_Funktionen" title="Konvexe und konkave Funktionen">Konvexität</a> oder <a href="Wohldefiniertheit" title="Wohldefiniertheit">Wohldefiniertheit</a>. Die erweiterte Funktion ist ebenfalls auf ganz <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span> definiert. Die Menge

</p>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {dom} ({\tilde {f}})=\{x\in V\mid {\tilde {f}}(x)<{+\infty }\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e73a1f0aedd12fb78eb5eec6d2f91dacc06ee126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.286ex; height:3.176ex;" alt="{\displaystyle \operatorname {dom} ({\tilde {f}})=\{x\in V\mid {\tilde {f}}(x)<{+\infty }\}}" loading="lazy"></span></dd></dl>

<p>heißt der <i>wesentliche Definitionsbereich</i> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6cb99679a4b79cb5ca3c242811bd91220c91f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}}" loading="lazy"></span>. Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dom} ({\tilde {f}})\neq \emptyset }">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {dom} ({\tilde {f}})\neq \emptyset }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c98943de6f0e694a78abfc7b12a32c763ab35bb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.16ex; height:3.176ex;" alt="{\displaystyle \operatorname {dom} ({\tilde {f}})\neq \emptyset }" loading="lazy"></span>, so heißt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6cb99679a4b79cb5ca3c242811bd91220c91f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.699ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}}" loading="lazy"></span> eine <i>echte Funktion</i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Monotonie">Monotonie</h3></div>

<p>Als Beispiel betrachten wir die Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }" loading="lazy"></span>, definiert durch <span class="mwe-math-element mwe-math-element-inline"><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)=-x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=-x^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/772f4a162f3b6f9325ef5f9404424969480dcd27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.708ex; height:3.176ex;" alt="{\displaystyle f(x)=-x^{2}}" loading="lazy"></span>. Sie ist <a href="Monoton_fallende_Funktion" class="mw-redirect" title="Monoton fallende Funktion">monoton fallend</a> auf dem Intervall <span class="mwe-math-element mwe-math-element-inline"><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,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8dc2d914c2df66bc0f7893bfb8da36766650fe47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}" loading="lazy"></span>. Um diese Eigenschaft nun auf ganz <span class="mwe-math-element mwe-math-element-inline"><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} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" 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} }" loading="lazy"></span> zu übertragen, setzen wir <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=[0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=[0,\infty )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9e38f8e2d61bde64da10f732d7757062779c2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.612ex; height:2.843ex;" alt="{\displaystyle M=[0,\infty )}" loading="lazy"></span>. Demnach gilt:

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<mtext> falls </mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mtext> sonst</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}(x)={\begin{cases}-x^{2}&{\text{ falls }}x\in [0,\infty )\\{+\infty }&{\text{ sonst}}\end{cases}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be5856b93ef34ba36f4ff98a036d69a11bb81ad6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.433ex; height:6.176ex;" alt="{\displaystyle {\tilde {f}}(x)={\begin{cases}-x^{2}&{\text{ falls }}x\in [0,\infty )\\{+\infty }&{\text{ sonst}}\end{cases}}}" loading="lazy"></span></dd></dl>

<p>Die erweiterte Funktion ist nun nach den <a href="Erweiterte_reelle_Zahl#Rechenregeln_aus_Stetigkeit" title="Erweiterte reelle Zahl">Rechenregeln mit unendlich</a> monoton fallend auf ganz <span class="mwe-math-element mwe-math-element-inline"><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} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" 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} }" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Konvexität"><span id="Konvexit.C3.A4t"></span>Konvexität</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" 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}" loading="lazy"></span> konvex auf der Menge <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span>, so ist die erweiterte konvexe Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}\colon V\to \mathbb {R} \cup \{{+\infty }\}}">

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}\colon V\to \mathbb {R} \cup \{{+\infty }\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/133aca60313606133baac0cc0f05097b95eb76a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.852ex; height:3.176ex;" alt="{\displaystyle {\tilde {f}}\colon V\to \mathbb {R} \cup \{{+\infty }\}}" loading="lazy"></span> durch

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> falls </mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mtext> sonst</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}(x)={\begin{cases}f(x)&{\text{ falls }}x\in C\\{+\infty }&{\text{ sonst}}\end{cases}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd3b5f6a5f450e7cd23150c8661b39380b8c2e26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.353ex; height:6.176ex;" alt="{\displaystyle {\tilde {f}}(x)={\begin{cases}f(x)&{\text{ falls }}x\in C\\{+\infty }&{\text{ sonst}}\end{cases}}}" loading="lazy"></span></dd></dl>

<p>definiert. Mit den Rechenregeln für unendlich ist diese Funktion nun konvex auf ganz <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" 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}" loading="lazy"></span> und nicht nur auf der Menge <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span>. Beispielsweise ist die Sinusfunktion konvex auf dem Intervall <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-\pi ,0]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [-\pi ,0]}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b78e1f27dbb353d8aa259a1ab97afe1a7e8a6cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.63ex; height:2.843ex;" alt="{\displaystyle [-\pi ,0]}" loading="lazy"></span>. Somit lautet die erweiterte Funktion

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> falls </mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mtext> sonst.</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}(x)={\begin{cases}\sin(x)&{\text{ falls }}x\in [-\pi ,0]\\{+\infty }&{\text{ sonst.}}\end{cases}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15fc1e40885055aab6b23060766ab55d0a9e0394.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.794ex; height:6.176ex;" alt="{\displaystyle {\tilde {f}}(x)={\begin{cases}\sin(x)&{\text{ falls }}x\in [-\pi ,0]\\{+\infty }&{\text{ sonst.}}\end{cases}}}" loading="lazy"></span></dd></dl>

<p>Diese Funktion ist nun konvex auf ganz <span class="mwe-math-element mwe-math-element-inline"><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} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" 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} }" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Definitionslücken"><span id="Definitionsl.C3.BCcken"></span>Definitionslücken</h3></div>

<p>Betrachtet man die Funktion <span class="mwe-math-element mwe-math-element-inline"><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)={\tfrac {1}{x^{2}}}}">

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\tfrac {1}{x^{2}}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e912acc95b3575a0a1e15d17650206da37bff04d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.124ex; height:3.843ex;" alt="{\displaystyle f(x)={\tfrac {1}{x^{2}}}}" loading="lazy"></span>, so ist diese an der Stelle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"></span> nicht definiert. Setzt man nun <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=D_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=D_{f}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c225e3bef7132c11ac31c60640c7013798df8ca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.601ex; height:2.843ex;" alt="{\displaystyle M=D_{f}}" loading="lazy"></span>, wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13b8420d715315c8e8b57f178a1b3a8512bd0959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.061ex; height:2.843ex;" alt="{\displaystyle D_{f}}" loading="lazy"></span> der Definitionsbereich ist, so gilt:

</p>

<mover>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mtext> falls </mtext>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

<mtd>

<mtext> sonst</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}(x)={\begin{cases}{+\infty }&{\text{ falls }}x=0\\{\frac {1}{x^{2}}}&{\text{ sonst}}\end{cases}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/334087ff20b7d9109e8c2e75aa6aaeb36f4a1fdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.722ex; height:6.676ex;" alt="{\displaystyle {\tilde {f}}(x)={\begin{cases}{+\infty }&{\text{ falls }}x=0\\{\frac {1}{x^{2}}}&{\text{ sonst}}\end{cases}}}" loading="lazy"></span></dd></dl>

<p>Die erweiterte Funktion ist jetzt auf ganz <span class="mwe-math-element mwe-math-element-inline"><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} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" 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} }" loading="lazy"></span> definiert und es können Operationen mit der Funktion ausgeführt werden, ohne Rücksicht auf die Definitionslücke zu nehmen. Es darf aber <i>nicht</i> aus der erweiterten Funktion geschlossen werden, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{0^{2}}}={+\infty }}">

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{0^{2}}}={+\infty }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/def01308d76ba5052953c63309ac557df52aac6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.72ex; height:4.009ex;" alt="{\displaystyle {\tfrac {1}{0^{2}}}={+\infty }}" loading="lazy"></span> gelte, da der Wert <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy"></span> erst im Nachhinein festgelegt wurde.

</p>

<div class="mw-heading mw-heading2"><h2 id="Verwendung">Verwendung</h2></div>

<p>Erweiterte Funktionen finden sich in vielen Bereichen der Analysis, insbesondere der <a href="Optimierung_(Mathematik)" class="mw-redirect" title="Optimierung (Mathematik)">Optimierung</a>. Hier bieten sie den Vorteil, dass man bei erweiterten Definitionen immer noch sinnvoll minimieren kann, aber keine formalen Probleme mit Definitionslücken oder nicht-konvexen Bereichen der Funktion bekommt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Carl Geiger, Christian Kanzow: <cite style="font-style:italic">Theorie und Numerik restringierter Optimierungsaufgaben</cite>. Springer-Verlag, Berlin Heidelberg New York 2002, ISBN 3-540-42790-2.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Erweiterte+Funktion&rft.au=Carl+Geiger%2C+Christian+Kanzow&rft.btitle=Theorie+und+Numerik+restringierter+Optimierungsaufgaben&rft.date=2002&rft.genre=book&rft.isbn=3540427902&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none"> </span></li>

<li>Stephen Boyd, Lieven Vandenberghe: <cite style="font-style:italic">Convex Optimization</cite>. Cambridge University Press, Cambridge, New York, Melbourne 2004, ISBN 978-0-521-83378-3 (<a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/">online</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Erweiterte+Funktion&rft.au=Stephen+Boyd%2C+Lieven+Vandenberghe&rft.btitle=Convex+Optimization&rft.date=2004&rft.genre=book&rft.isbn=9780521833783&rft.place=Cambridge%2C+New+York%2C+Melbourne&rft.pub=Cambridge+University+Press" style="display:none"> </span></li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2021-05-13" href="https://de.wikipedia.org/wiki/?title=Erweiterte_Funktion&oldid=211907808">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>