GermanWikipedia - /page/entry.mu

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<title>Halbstetigkeit</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Halbstetigkeit</h1>

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<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">In der <a href="Mathematik" title="Mathematik">Mathematik</a> heißt eine <a href="Reellwertige_Funktion" title="Reellwertige Funktion">reellwertige Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><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"> oberhalbstetig (oder halbstetig von oben) in einem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy">, wenn die Funktionswerte für <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Argumente</a> nahe bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> ausgehend nicht nach oben springen. Wenn die Funktionswerte nicht nach unten springen, dann heißt die Funktion unterhalbstetig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> (oder halbstetig von unten).

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

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"> ein <a href="Topologischer_Raum" title="Topologischer Raum">topologischer Raum</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to \mathbb {R} }">

</mrow>

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359ea801448b482438cb2149cfce6559dc3385b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.585ex; height:2.509ex;" alt="{\displaystyle f\colon X\to \mathbb {R} }" loading="lazy"> eine reellwertige Funktion. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><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"> heißt in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> oberhalbstetig, wenn für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> eine Umgebung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> existiert, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)<f(x_{0})+\varepsilon }">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(y)<f(x_{0})+\varepsilon }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54b392fc00c190792fea9c23df775979ab5b00f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.738ex; height:2.843ex;" alt="{\displaystyle f(y)<f(x_{0})+\varepsilon }" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> gilt. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"> ein Raum, in dem jede folgenstetige Funktion auch stetig ist, etwa ein <a href="Metrischer_Raum" title="Metrischer Raum">metrischer Raum</a>, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> genau dann oberhalbstetig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy">, falls

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle \limsup _{y\to x}f(y)\leq f(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9edaa1636d96f0a6452a3ad1ce42e66b0c9c722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.265ex; height:4.676ex;" alt="{\displaystyle \limsup _{y\to x}f(y)\leq f(x)}" loading="lazy">.</dd></dl>

</mstyle>

</mrow>

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

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</math><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"> heißt oberhalbstetig auf einer Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy">, wenn sie in jedem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ffbdb59406dc64aa6769cecf0e9ee109d181119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.509ex;" alt="{\displaystyle x_{0}\in M}" loading="lazy"> oberhalbstetig ist. Ist dabei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> der ganze topologische Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy">, so heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> oberhalbstetig.

Analog heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> unterhalbstetig, wenn für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> eine Umgebung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> existiert, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)>f(x_{0})-\varepsilon }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(y)>f(x_{0})-\varepsilon }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6cdd6ef002d417a3ae5766177a87ebdfa52bf40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.738ex; height:2.843ex;" alt="{\displaystyle f(y)>f(x_{0})-\varepsilon }" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"> ein Raum, in dem jede folgenstetige Funktion auch stetig ist, etwa ein metrischer Raum, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> genau dann unterhalbstetig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy">, falls

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \liminf _{y\to x}f(y)\geq f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ef32e1c8ab79c1417f302bb48353981f26324df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.415ex; height:4.176ex;" alt="{\displaystyle \liminf _{y\to x}f(y)\geq f(x)}" loading="lazy">.</dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math><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"> heißt unterhalbstetig auf einer Teilmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy">, wenn sie in jedem Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ffbdb59406dc64aa6769cecf0e9ee109d181119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.509ex;" alt="{\displaystyle x_{0}\in M}" loading="lazy"> unterhalbstetig ist. Ist dabei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> der ganze topologische Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy">, so heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> unterhalbstetig.

Zusammenhang der beiden Halbstetigkeitsbegriffe: Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ist genau dann oberhalbstetig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b79e955b57dd7aada93b8afd459996ae941d480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X}" loading="lazy"> bzw. auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\subseteq X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf429fb33f3cfe051b0f7ade626d238a65a7473b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.521ex; height:2.343ex;" alt="{\displaystyle M\subseteq X}" loading="lazy"> wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0edfedee3fca0a26dd6f515e7ed9517a4e2cd04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.087ex; height:2.509ex;" alt="{\displaystyle -f}" loading="lazy"> unterhalbstetig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in X}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b79e955b57dd7aada93b8afd459996ae941d480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.205ex; height:2.509ex;" alt="{\displaystyle x_{0}\in X}" loading="lazy"> bzw. auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\subseteq X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf429fb33f3cfe051b0f7ade626d238a65a7473b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.521ex; height:2.343ex;" alt="{\displaystyle M\subseteq X}" loading="lazy"> ist.

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

Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf85883d74b75fe35ca8d3f2b44802df078e4fa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=0}" loading="lazy"> für <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a4dbbf970b2d2863dcab589eafe006f08e727d7.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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea78f54e69b72f398cf6077e61c50a05b532d4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=1}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2608e2b392b079f5b763f27bf52883dbee3b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}" loading="lazy"> ist oberhalbstetig, aber nicht unterhalbstetig in <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><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">.

Denn entfernt man sich mit den Argumenten in negative Richtung von der 0, dann springen die Funktionswerte plötzlich von 1 auf 0, aber sie springen nicht nach oben, egal wohin man sich entfernt.

Die <a href="Gau%C3%9Fklammer" class="mw-redirect" title="Gaußklammer">Gaußklammer</a> ist oberhalbstetig, denn sie verhält sich an jeder ganzen Zahl so wie die eben beschriebene Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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">.

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

Eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ist <a href="Stetige_Funktion" title="Stetige Funktion">stetig</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> genau dann, wenn sie dort halbstetig von oben und von unten ist. Dies folgt aus einer Umformulierung der Stetigkeitsbedingung, welche lautet, dass es zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> gibt, sodass für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> aus der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy">-Umgebung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> die Ungleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)-f(x_{0})|<\varepsilon }">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f(x)-f(x_{0})|<\varepsilon }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c85be121472f6c05895e45cbbdcdbe0b2d4e6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.205ex; height:2.843ex;" alt="{\displaystyle |f(x)-f(x_{0})|<\varepsilon }" loading="lazy"> gilt. Diese ist äquivalent zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\varepsilon \leq f(x)-f(x_{0})\leq \varepsilon }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\varepsilon \leq f(x)-f(x_{0})\leq \varepsilon }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87aca9295bad6002a91e5e307f3c589de7348cde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.902ex; height:2.843ex;" alt="{\displaystyle -\varepsilon \leq f(x)-f(x_{0})\leq \varepsilon }" loading="lazy">. Hier bedeutet das linke Ungleichheitszeichen die untere Halbstetigkeit und das rechte Ungleichheitszeichen die obere Halbstetigkeit.

Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> zwei in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> oberhalbstetige Funktionen, dann ist auch ihre Summe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d94a24abd865f6f9fd67a7df7e531cae1c769b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.235ex; height:2.509ex;" alt="{\displaystyle f+g}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> oberhalbstetig. Sind beide Funktionen nichtnegativ in einer Umgebung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy">, dann ist auch das Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06bac4638bb56f14688118ce88c188c7a021eb29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> oberhalbstetig. Die Multiplikation einer positiven oberhalbstetigen Funktion mit einer negativen <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahl</a> ergibt eine unterhalbstetige Funktion.

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> eine <a href="Kompakter_Raum" title="Kompakter Raum">kompakte Menge</a> (zum Beispiel ein <a href="Abgeschlossenes_Intervall" class="mw-redirect" title="Abgeschlossenes Intervall">abgeschlossenes Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" 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]}" loading="lazy"> mit reellen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91a7698e4c7401bb321f97888b872b583a9e4642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}" loading="lazy">) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon D\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229e3a78dd4be03159691c3d4c84a9095784d26e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.529ex; height:2.509ex;" alt="{\displaystyle f\colon D\to \mathbb {R} }" loading="lazy"> oberhalbstetig, dann hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein Maximum auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">. Analoges gilt für eine unterhalbstetige Funktion und ihr Minimum.

Sind die Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}\colon X\to \mathbb {R} }">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70dc0cc61a02051e1349f8172775443730d9542b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.664ex; height:2.509ex;" alt="{\displaystyle f_{n}\colon X\to \mathbb {R} }" loading="lazy"> (für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy">) unterhalbstetig und ihr <a href="Supremum" class="mw-redirect" title="Supremum">Supremum</a>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x):=\sup\{f_{n}(x):n\in \mathbb {N} \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d98cce75e4054b47db9334c1eb3a23ce7283e71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.336ex; height:2.843ex;" alt="{\displaystyle f(x):=\sup\{f_{n}(x):n\in \mathbb {N} \}}" loading="lazy"></dd></dl>

kleiner als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy"> für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy">, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> unterhalbstetig. Selbst wenn alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"> stetig sind, muss <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> aber nicht stetig sein.

<div class="mw-heading mw-heading2"><h2 id="Alternative_Beschreibung">Alternative Beschreibung</h2></div>

Durch eine geeignete Wahl einer <a href="Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a> auf <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><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"> können oberhalbstetige und unterhalbstetige Funktionen als stetige Funktionen aufgefasst werden, und somit lassen sich einige der Eigenschaften direkt aus allgemeinen Aussagen aus der Topologie herleiten.

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{<}:={\Big \{}\left]-\infty ,a\right[;a\in \mathbb {R} \cup \{+\infty \}{\Big \}}\cup \{\varnothing \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a29c0ea170ba9217eb27d26f13bf463a1625558.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.131ex; height:4.843ex;" alt="{\displaystyle O_{<}:={\Big \{}\left]-\infty ,a\right[;a\in \mathbb {R} \cup \{+\infty \}{\Big \}}\cup \{\varnothing \}}" loading="lazy"> ist eine Topologie auf <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><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">. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,O)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,O)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da206bd2cd484aa06f5d10b7416d82a416c5b2cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle (X,O)}" loading="lazy"> ein topologischer Raum. Eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359ea801448b482438cb2149cfce6559dc3385b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.585ex; height:2.509ex;" alt="{\displaystyle f\colon X\to \mathbb {R} }" loading="lazy"> ist genau dann oberhalbstetig, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> als Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,O)\to (\mathbb {R} ,O_{<})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,O)\to (\mathbb {R} ,O_{<})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e314ba5a4d7f6d909fecfc3faae9a9411d929e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.016ex; height:2.843ex;" alt="{\displaystyle (X,O)\to (\mathbb {R} ,O_{<})}" loading="lazy"> stetig ist.

Für unterhalbstetige Funktionen verwendet man analog die Topologie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{>}:={\Big \{}\left]a,\infty \right[;a\in \mathbb {R} \cup \{-\infty \}{\Big \}}\cup \{\varnothing \}}">

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{>}:={\Big \{}\left]a,\infty \right[;a\in \mathbb {R} \cup \{-\infty \}{\Big \}}\cup \{\varnothing \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0cce2cdefe1856be02c303e08fe845d1bcd1196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.323ex; height:4.843ex;" alt="{\displaystyle O_{>}:={\Big \{}\left]a,\infty \right[;a\in \mathbb {R} \cup \{-\infty \}{\Big \}}\cup \{\varnothing \}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Schwach_halbstetige_Funktionen">Schwach halbstetige Funktionen</h2></div>

Eine Verallgemeinerung der halbstetigen Funktionen sind die schwach halbstetigen Funktionen. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"> ein <a href="Normierter_Raum" title="Normierter Raum">normierter Raum</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subset X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/616c8a284e5623c9ca7184dddef9b84da4cd251b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.578ex; height:2.176ex;" alt="{\displaystyle S\subset X}" loading="lazy"> eine Teilmenge. Eine Funktion oder ein Funktional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\supset S\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0e1c843d951024cccb32d70c8eae8d2cd250817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.182ex; height:2.509ex;" alt="{\displaystyle f\colon X\supset S\to \mathbb {R} }" loading="lazy"> heißt

<ul><li>schwach unterhalbstetig auf der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, wenn für jede Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})_{n\in \mathbb {N} }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n})_{n\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ebf5fbc73781898c4395888facf8a8ef83a7cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.859ex; height:2.843ex;" alt="{\displaystyle (x_{n})_{n\in \mathbb {N} }}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, die <a href="Schwache_Konvergenz" title="Schwache Konvergenz">schwach gegen</a> ihren schwachen Grenzwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}\in S}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34b985b61852129b84484fec7c03047f277c99cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle {\tilde {x}}\in S}" loading="lazy"> konvergiert, gilt, dass</li></ul>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \liminf _{n\to \infty }f(x_{n})\geq f({\tilde {x}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e62ba4c3e1da3bb801de491659aa23ca15810bc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.808ex; height:3.843ex;" alt="{\displaystyle \liminf _{n\to \infty }f(x_{n})\geq f({\tilde {x}})}" loading="lazy">.</dd></dl>

<ul><li>schwach oberhalbstetig auf der Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, wenn für jede Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})_{n\in \mathbb {N} }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n})_{n\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ebf5fbc73781898c4395888facf8a8ef83a7cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.859ex; height:2.843ex;" alt="{\displaystyle (x_{n})_{n\in \mathbb {N} }}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, die schwach gegen ihren schwachen Grenzwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}\in S}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34b985b61852129b84484fec7c03047f277c99cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle {\tilde {x}}\in S}" loading="lazy"> konvergiert, gilt, dass</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \limsup _{n\to \infty }f(x_{n})\leq f({\tilde {x}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ad2c9f504ee6d0b055286142a84abbecbdbe065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.658ex; height:4.343ex;" alt="{\displaystyle \limsup _{n\to \infty }f(x_{n})\leq f({\tilde {x}})}" loading="lazy">.</dd></dl>

Beispielsweise sind stetige <a href="Quasikonvexe_Funktion" title="Quasikonvexe Funktion">quasikonvexe Funktionen</a> schwach unterhalbstetig. Äquivalent zur schwachen Unterhalbstetigkeit einer Funktion ist, dass ihr <a href="Epigraph_(Mathematik)" title="Epigraph (Mathematik)">Epigraph</a> eine <a href="Schwach_folgenabgeschlossene_Menge" title="Schwach folgenabgeschlossene Menge">schwach folgenabgeschlossene Menge</a> ist. Schwach unterhalbstetige Funktionen spielen eine wichtige Rolle in der Optimierung, da sie auf <a href="Schwach_folgenkompakte_Menge" title="Schwach folgenkompakte Menge">schwach folgenkompakten Mengen</a> immer ein Minimum annehmen.

<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:Halbstetigkeit&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"> </li></ul></div><div><div class="zim-footer">

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