SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Semi-continuity</title>

</head>

Semi-continuity

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">For the notion of upper or lower semi-continuous <a href="Set-valued_function" title="Set-valued function">set-valued function</a>, see <a href="Hemicontinuity" title="Hemicontinuity">Hemicontinuity</a>.</div>

In <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, semicontinuity (or semi-continuity) is a property of <a href="Extended_real_number" class="mw-redirect" title="Extended real number">extended real</a>-valued <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> that is weaker than <a href="Continuous_function" title="Continuous function">continuity</a>. An extended real-valued function <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="./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"> is upper (respectively, lower) semicontinuous at a point <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="./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"> if, roughly speaking, the function values for arguments near <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="./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"> are not much higher (respectively, lower) than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x_{0}\right).}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c29828810d533ec984004e10a0b54690549e975e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.893ex; height:2.843ex;" alt="{\displaystyle f\left(x_{0}\right).}" loading="lazy"> Briefly, a function on a domain <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="./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"> is lower semi-continuous if its <a href="Epigraph_(mathematics)" title="Epigraph (mathematics)">epigraph</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}</annotation>

</semantics>

</math><img src="./ef87535151cc17f62b98d99e1736bc4961f3deab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.969ex; height:2.843ex;" alt="{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}" loading="lazy"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./65a49d46b78287766b5b1ae2ca218941b7cb1fa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.498ex; height:2.176ex;" alt="{\displaystyle X\times \mathbb {R} }" loading="lazy">, and upper semi-continuous if <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="./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"> is lower semi-continuous.

A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point <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="./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"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x_{0}\right)+c}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./db3ba208d6d4ffb9d15eafec9d22131a424fcedd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.706ex; height:2.843ex;" alt="{\displaystyle f\left(x_{0}\right)+c}" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2ba126f626d61752f62eaacaf11761a54de4dc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}" loading="lazy">, then the result is upper semicontinuous; if we decrease its value to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x_{0}\right)-c}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./14522580f57f284bd3ca3df4201810244225beb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.706ex; height:2.843ex;" alt="{\displaystyle f\left(x_{0}\right)-c}" loading="lazy"> then the result is lower semicontinuous.

The notion of upper and lower semicontinuous function was first introduced and studied by <a href="Ren%C3%A9_Baire" class="mw-redirect" title="René Baire">René Baire</a> in his thesis in 1899.<a href="#cite_note-1">[1]</a>

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

Assume throughout that <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="./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"> is a <a href="Topological_space" title="Topological space">topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./809ed19a0ef844995d674e77acd2f0c8b03a3d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.603ex; height:3.343ex;" alt="{\displaystyle f:X\to {\overline {\mathbb {R} }}}" loading="lazy"> is a function with values in the <a href="Extended_real_number" class="mw-redirect" title="Extended real number">extended real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}=[-\infty ,\infty ]}">

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}=[-\infty ,\infty ]}</annotation>

</semantics>

</math><img src="./b095388ff2cb62df5a45671a0d2dae6dd9ff4db9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.848ex; height:3.509ex;" alt="{\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}=[-\infty ,\infty ]}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Upper_semicontinuity">Upper semicontinuity</h3></div>

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./809ed19a0ef844995d674e77acd2f0c8b03a3d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.603ex; height:3.343ex;" alt="{\displaystyle f:X\to {\overline {\mathbb {R} }}}" loading="lazy"> is called upper semicontinuous at a point <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="./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"> if for every real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y>f\left(x_{0}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1a29fe445e064d17fa30b7a9fb468a079a7333ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.113ex; height:2.843ex;" alt="{\displaystyle y>f\left(x_{0}\right)}" loading="lazy"> there exists a <a href="Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> <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="./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"> of <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="./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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)<y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a4fa36f504cefd75250a660100a90e767692fd04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle f(x)<y}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c32ddcb2941216f2980b950ce969dc15cba26906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}" loading="lazy">.<a href="#cite_note-Stromberg-2">[2]</a>

Equivalently, <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="./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"> is upper semicontinuous at <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="./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"> if and only if

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math>

where lim sup is the <a href="Limit_superior_(topological_space)" class="mw-redirect" title="Limit superior (topological space)">limit superior</a> of the function <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="./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"> at the point <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="./7ec756993d89cc1bd74f84040c07f5e11f0a8102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0}.}" loading="lazy">

Here the limit superior is defined as

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \inf _{U\ni x_{0}}\sup _{x\in U}f(x)}</annotation>

</semantics>

</math>

where the supremum is taken over all neighborhoods of <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="./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">.

If <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="./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"> is a <a href="Metric_space" title="Metric space">metric space</a> with <a href="Distance_function" class="mw-redirect" title="Distance function">distance function</a> <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="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0})\in \mathbb {R} ,}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{0})\in \mathbb {R} ,}</annotation>

</semantics>

</math><img src="./8927612a1fb098eb5dabe0edfe25bf9f6b570d44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.637ex; height:2.843ex;" alt="{\displaystyle f(x_{0})\in \mathbb {R} ,}" loading="lazy"> this can also be restated using an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">-<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="./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"> formulation, similar to the definition of <a href="Continuous_function" title="Continuous function">continuous function</a>. Namely, for each <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="./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"> there is a <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="./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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)<f(x_{0})+\varepsilon }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ae63f6b64357ec1e97418ba8e66bcf8f3c7d3dbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.912ex; height:2.843ex;" alt="{\displaystyle f(x)<f(x_{0})+\varepsilon }" loading="lazy"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x_{0})<\delta .}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d(x,x_{0})<\delta .}</annotation>

</semantics>

</math><img src="./c258afa8d6675f4b79c8fc96a443cf2ae32b8ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.567ex; height:2.843ex;" alt="{\displaystyle d(x,x_{0})<\delta .}" loading="lazy">

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./809ed19a0ef844995d674e77acd2f0c8b03a3d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.603ex; height:3.343ex;" alt="{\displaystyle f:X\to {\overline {\mathbb {R} }}}" loading="lazy"> is called upper semicontinuous if it satisfies any of the following equivalent conditions:<a href="#cite_note-Stromberg-2">[2]</a>

<dl><dd>(1) The function is upper semicontinuous at every point of its <a href="Domain_(function)" class="mw-redirect" title="Domain (function)">domain</a>.</dd>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e17075a7abdf541b17004d8b9d0dac081860d685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.674ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {R} }" loading="lazy">, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}([-\infty ,y))=\{x\in X:f(x)<y\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}([-\infty ,y))=\{x\in X:f(x)<y\}}</annotation>

</semantics>

</math><img src="./c74f035e627a6951ed9311810d2f4f453c2735d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.518ex; height:3.176ex;" alt="{\displaystyle f^{-1}([-\infty ,y))=\{x\in X:f(x)<y\}}" loading="lazy"> is <a href="Open_(topology)" class="mw-redirect" title="Open (topology)">open</a> 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="./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">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-\infty ,y)=\{t\in {\overline {\mathbb {R} }}:t<y\}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [-\infty ,y)=\{t\in {\overline {\mathbb {R} }}:t<y\}}</annotation>

</semantics>

</math><img src="./43330d6dc212481fe03635109d0a60bb51c9b2ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.8ex; height:3.509ex;" alt="{\displaystyle [-\infty ,y)=\{t\in {\overline {\mathbb {R} }}:t<y\}}" loading="lazy">.</dd>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e17075a7abdf541b17004d8b9d0dac081860d685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.674ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {R} }" loading="lazy">, the <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="./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">-<a href="Superlevel_set" class="mw-redirect" title="Superlevel set">superlevel set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}([y,\infty ))=\{x\in X:f(x)\geq y\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}([y,\infty ))=\{x\in X:f(x)\geq y\}}</annotation>

</semantics>

</math><img src="./57678383a7aa6df5e809304fe881c1f1285e1199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.71ex; height:3.176ex;" alt="{\displaystyle f^{-1}([y,\infty ))=\{x\in X:f(x)\geq y\}}" loading="lazy"> is <a href="Closed_(topology)" class="mw-redirect" title="Closed (topology)">closed</a> 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="./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">.</dd>

<dd>(4) The <a href="Hypograph_(mathematics)" title="Hypograph (mathematics)">hypograph</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\leq f(x)\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\leq f(x)\}}</annotation>

</semantics>

</math><img src="./ec81bab21f675f9c5680a4c562ac2082a21d198e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.969ex; height:2.843ex;" alt="{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\leq f(x)\}}" loading="lazy"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./65a49d46b78287766b5b1ae2ca218941b7cb1fa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.498ex; height:2.176ex;" alt="{\displaystyle X\times \mathbb {R} }" loading="lazy">.</dd>

<dd>(5) The function <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="./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"> is continuous when the <a href="Codomain" title="Codomain">codomain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c62ecdf73d36e9d05a018732fd5f51db73d0c55b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.793ex; height:3.009ex;" alt="{\displaystyle {\overline {\mathbb {R} }}}" loading="lazy"> is given the <a href="Left_order_topology" class="mw-redirect" title="Left order topology">left order topology</a>. This is just a restatement of condition (2) since the left order topology is generated by all the intervals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-\infty ,y)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e738e065b6d2479c0749ee4d2d9bbdb4bb99a614.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.873ex; height:2.843ex;" alt="{\displaystyle [-\infty ,y)}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Lower_semicontinuity">Lower semicontinuity</h3></div>

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./809ed19a0ef844995d674e77acd2f0c8b03a3d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.603ex; height:3.343ex;" alt="{\displaystyle f:X\to {\overline {\mathbb {R} }}}" loading="lazy"> is called lower semicontinuous at a point <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="./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"> if for every real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<f\left(x_{0}\right)}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3bad634041bbf2b8807fe47903fa79bd5e99c6c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.113ex; height:2.843ex;" alt="{\displaystyle y<f\left(x_{0}\right)}" loading="lazy"> there exists a <a href="Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> <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="./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"> of <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="./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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)>y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5e214848bc216b97ceff4740b8f773609136f696.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle f(x)>y}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c32ddcb2941216f2980b950ce969dc15cba26906.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}" loading="lazy">.

Equivalently, <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="./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"> is lower semicontinuous at <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="./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"> if and only if

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \liminf _{x\to x_{0}}f(x)\geq f(x_{0})}</annotation>

</semantics>

</math>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d53498a9f403b1cf8cb38888e0c312a790c190b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.153ex; width:6.422ex; height:2.176ex;" alt="{\displaystyle \liminf }" loading="lazy"> is the <a href="Limit_inferior_(topological_space)" class="mw-redirect" title="Limit inferior (topological space)">limit inferior</a> of the function <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="./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"> at point <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="./7ec756993d89cc1bd74f84040c07f5e11f0a8102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0}.}" loading="lazy">

If <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="./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"> is a <a href="Metric_space" title="Metric space">metric space</a> with <a href="Distance_function" class="mw-redirect" title="Distance function">distance function</a> <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="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0})\in \mathbb {R} ,}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{0})\in \mathbb {R} ,}</annotation>

</semantics>

</math><img src="./8927612a1fb098eb5dabe0edfe25bf9f6b570d44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.637ex; height:2.843ex;" alt="{\displaystyle f(x_{0})\in \mathbb {R} ,}" loading="lazy"> this can also be restated as follows: For each <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="./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"> there is a <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="./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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)>f(x_{0})-\varepsilon }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2ac8bf72e41af82cbdca9d86f0cfdf756df81110.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.912ex; height:2.843ex;" alt="{\displaystyle f(x)>f(x_{0})-\varepsilon }" loading="lazy"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x_{0})<\delta .}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d(x,x_{0})<\delta .}</annotation>

</semantics>

</math><img src="./c258afa8d6675f4b79c8fc96a443cf2ae32b8ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.567ex; height:2.843ex;" alt="{\displaystyle d(x,x_{0})<\delta .}" loading="lazy">

A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./809ed19a0ef844995d674e77acd2f0c8b03a3d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.603ex; height:3.343ex;" alt="{\displaystyle f:X\to {\overline {\mathbb {R} }}}" loading="lazy"> is called lower semicontinuous if it satisfies any of the following equivalent conditions:

<dl><dd>(1) The function is lower semicontinuous at every point of its <a href="Domain_(function)" class="mw-redirect" title="Domain (function)">domain</a>.</dd>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e17075a7abdf541b17004d8b9d0dac081860d685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.674ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {R} }" loading="lazy">, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}((y,\infty ])=\{x\in X:f(x)>y\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}((y,\infty ])=\{x\in X:f(x)>y\}}</annotation>

</semantics>

</math><img src="./272409a931166756ce2b7a7f5d61addf37debcf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.71ex; height:3.176ex;" alt="{\displaystyle f^{-1}((y,\infty ])=\{x\in X:f(x)>y\}}" loading="lazy"> is <a href="Open_(topology)" class="mw-redirect" title="Open (topology)">open</a> 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="./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">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y,\infty ]=\{t\in {\overline {\mathbb {R} }}:t>y\}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y,\infty ]=\{t\in {\overline {\mathbb {R} }}:t>y\}}</annotation>

</semantics>

</math><img src="./d11d06cc7126d67bbd5f8d5ca4d0386d386b5d1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.992ex; height:3.509ex;" alt="{\displaystyle (y,\infty ]=\{t\in {\overline {\mathbb {R} }}:t>y\}}" loading="lazy">.</dd>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e17075a7abdf541b17004d8b9d0dac081860d685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.674ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {R} }" loading="lazy">, the <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="./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">-<a href="Sublevel_set" class="mw-redirect" title="Sublevel set">sublevel set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}((-\infty ,y])=\{x\in X:f(x)\leq y\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}((-\infty ,y])=\{x\in X:f(x)\leq y\}}</annotation>

</semantics>

</math><img src="./ccf04b7f5717be347b1db24d9fe217381b32aeb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.518ex; height:3.176ex;" alt="{\displaystyle f^{-1}((-\infty ,y])=\{x\in X:f(x)\leq y\}}" loading="lazy"> is <a href="Closed_(topology)" class="mw-redirect" title="Closed (topology)">closed</a> 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="./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">.</dd>

<dd>(4) The <a href="Epigraph_(mathematics)" title="Epigraph (mathematics)">epigraph</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}</annotation>

</semantics>

</math><img src="./ef87535151cc17f62b98d99e1736bc4961f3deab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.969ex; height:2.843ex;" alt="{\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}" loading="lazy"> is closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./65a49d46b78287766b5b1ae2ca218941b7cb1fa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.498ex; height:2.176ex;" alt="{\displaystyle X\times \mathbb {R} }" loading="lazy">.<a href="#cite_note-Kurdila2005-3">[3]</a>: 207 </dd>

<dd>(5) The function <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="./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"> is continuous when the <a href="Codomain" title="Codomain">codomain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c62ecdf73d36e9d05a018732fd5f51db73d0c55b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.793ex; height:3.009ex;" alt="{\displaystyle {\overline {\mathbb {R} }}}" loading="lazy"> is given the <a href="Right_order_topology" class="mw-redirect" title="Right order topology">right order topology</a>. This is just a restatement of condition (2) since the right order topology is generated by all the intervals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y,\infty ]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1a68e44010ee6b3e4ce49e177dc9bb343833a799.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.065ex; height:2.843ex;" alt="{\displaystyle (y,\infty ]}" loading="lazy">.</dd></dl>

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

Consider the function <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="./9e9687ea22c0f310582e97ee5f6c6a5fca28203d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f,}" loading="lazy"> <a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> defined by:

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}-1&{\mbox{if }}x<0,\\1&{\mbox{if }}x\geq 0\end{cases}}}</annotation>

</semantics>

</math>

This function is upper semicontinuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=0,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a72c7ee7f5880074db8a515f79734b09160980b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.292ex; height:2.509ex;" alt="{\displaystyle x_{0}=0,}" loading="lazy"> but not lower semicontinuous.

The <a href="Floor_function" class="mw-redirect" title="Floor function">floor function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\lfloor x\rfloor ,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./190c830a762b949dfa2739040a4a55551338d2ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.557ex; height:2.843ex;" alt="{\displaystyle f(x)=\lfloor x\rfloor ,}" loading="lazy"> which returns the greatest integer less than or equal to a given real number <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="./feff4d40084c7351bf57b11ba2427f6331f5bdbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}" loading="lazy"> is everywhere upper semicontinuous. Similarly, the <a href="Ceiling_function" class="mw-redirect" title="Ceiling function">ceiling function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\lceil x\rceil }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c217512fdb5feebcc2d2c07fe8f55a9174cbaab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.91ex; height:2.843ex;" alt="{\displaystyle f(x)=\lceil x\rceil }" loading="lazy"> is lower semicontinuous.

Upper and lower semicontinuity bear no relation to <a href="Continuous_function#Directional_and_semi-continuity" title="Continuous function">continuity from the left or from the right</a> for functions of a real variable. Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain.<a href="#cite_note-4">[4]</a> For example the function

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}\sin(1/x)&{\mbox{if }}x\neq 0,\\1&{\mbox{if }}x=0,\end{cases}}}</annotation>

</semantics>

</math>

is upper semicontinuous at <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="./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"> while the function limits from the left or right at zero do not even exist.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./16344851ca00c2c8c6c80f658d1a180eb94abee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.975ex; height:2.343ex;" alt="{\displaystyle X=\mathbb {R} ^{n}}" loading="lazy"> is a Euclidean space (or more generally, a metric space) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =C([0,1],X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma =C([0,1],X)}</annotation>

</semantics>

</math><img src="./ec5e166b2c51fd7a377e58ccb71fd1f445fc7732.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.793ex; height:2.843ex;" alt="{\displaystyle \Gamma =C([0,1],X)}" loading="lazy"> is the space of <a href="Curve" title="Curve">curves</a> 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="./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"> (with the <a href="Supremum_norm" class="mw-redirect" title="Supremum norm">supremum distance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\Gamma }(\alpha ,\beta )=\sup\{d_{X}(\alpha (t),\beta (t)):t\in [0,1]\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{\Gamma }(\alpha ,\beta )=\sup\{d_{X}(\alpha (t),\beta (t)):t\in [0,1]\}}</annotation>

</semantics>

</math><img src="./70a1f48434a8ad4fece85cd420bf0ddbdc767a4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.128ex; height:2.843ex;" alt="{\displaystyle d_{\Gamma }(\alpha ,\beta )=\sup\{d_{X}(\alpha (t),\beta (t)):t\in [0,1]\}}" loading="lazy">), then the length functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L:\Gamma \to [0,+\infty ],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L:\Gamma \to [0,+\infty ],}</annotation>

</semantics>

</math><img src="./9ae1609c7c11f3aa1f8de31b212f52fb0ed747a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.856ex; height:2.843ex;" alt="{\displaystyle L:\Gamma \to [0,+\infty ],}" loading="lazy"> which assigns to each curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> its <a href="Curve#Length_of_curves" title="Curve">length</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\alpha ),}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4028f0ba4f8992b8af49ddd50f373d90f045b216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.527ex; height:2.843ex;" alt="{\displaystyle L(\alpha ),}" loading="lazy"> is lower semicontinuous.<a href="#cite_note-5">[5]</a> As an example, consider approximating the unit square diagonal by a staircase from below. The staircase always has length 2, while the diagonal line has only length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation>

</semantics>

</math><img src="./b4afc1e27d418021bf10898eb44a7f5f315735ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" loading="lazy">.

A fundamental example in <a href="Real_analysis" title="Real analysis">real analysis</a> is <a href="Fatou's_lemma" title="Fatou's lemma">Fatou's lemma</a>. It asserts that if <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="./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"> is a sequence of non-negative <a href="Measurable_function" title="Measurable function">measurable functions</a>, then

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d53498a9f403b1cf8cb38888e0c312a790c190b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.153ex; width:6.422ex; height:2.176ex;" alt="{\displaystyle \liminf }" loading="lazy"> denotes the (<a href="Pointwise" title="Pointwise">pointwise</a>) <a href="Limit_inferior" class="mw-redirect" title="Limit inferior">limit inferior</a>. What this means, in full generality, is that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\mu )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2683d38fc1ddf9d1933f7da7cfe773a1582ea15b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.225ex; height:2.843ex;" alt="{\displaystyle (X,\mu )}" loading="lazy"> be a measure space and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{+}(X,\mu )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{+}(X,\mu )}</annotation>

</semantics>

</math><img src="./14876d40a019f101d5627ee74355499372925055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.319ex; height:3.009ex;" alt="{\displaystyle L^{+}(X,\mu )}" loading="lazy"> denotes the set of positive measurable functions endowed with the topology of <a href="Convergence_in_measure" title="Convergence in measure">convergence in measure</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1e7e1ef161a49a22b500d63307460ad92eeb6a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.049ex; height:2.176ex;" alt="{\displaystyle \mu ,}" loading="lazy"> then the integral, seen as an operator from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{+}(X,\mu )}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{+}(X,\mu )}</annotation>

</semantics>

</math><img src="./14876d40a019f101d5627ee74355499372925055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.319ex; height:3.009ex;" alt="{\displaystyle L^{+}(X,\mu )}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-\infty ,+\infty ]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a7a5a5ba47896e8fdd43cf7f311db27350f0f8ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.591ex; height:2.843ex;" alt="{\displaystyle [-\infty ,+\infty ]}" loading="lazy"> is lower semicontinuous.

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

Unless specified otherwise, all functions below are from a <a href="Topological_space" title="Topological space">topological space</a> <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="./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"> to the <a href="Extended_real_number" class="mw-redirect" title="Extended real number">extended real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {R} }}=[-\infty ,\infty ].}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {R} }}=[-\infty ,\infty ].}</annotation>

</semantics>

</math><img src="./c99d0218a0b787a96d52c204201e1c27f520914a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.322ex; height:3.509ex;" alt="{\displaystyle {\overline {\mathbb {R} }}=[-\infty ,\infty ].}" loading="lazy"> Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated for semicontinuity over the whole domain.

<ul><li>A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./809ed19a0ef844995d674e77acd2f0c8b03a3d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.603ex; height:3.343ex;" alt="{\displaystyle f:X\to {\overline {\mathbb {R} }}}" loading="lazy"> is <a href="Continuous_function" title="Continuous function">continuous</a> if and only if it is both upper and lower semicontinuous.</li>

<li>The <a href="Indicator_function" title="Indicator function">characteristic function</a> or <a href="Indicator_function" title="Indicator function">indicator function</a> of a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./826569be03f873b81cdc6f12637ef5520c369d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.822ex; height:2.176ex;" alt="{\displaystyle A\subset X}" loading="lazy"> (defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{A}(x)=1}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9394f68a77dda0cf344969bc54ab234bb18f3ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.201ex; height:2.843ex;" alt="{\displaystyle \mathbf {1} _{A}(x)=1}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27bcc9b2afb295d4234bc294860cd0c63bcad2ca.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 x\in A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\notin A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8152431575305d6a6145adf9b279891a65923eba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.913ex; height:2.676ex;" alt="{\displaystyle x\notin A}" loading="lazy">) is upper semicontinuous if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a <a href="Closed_set" title="Closed set">closed set</a>. It is lower semicontinuous if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is an <a href="Open_set" title="Open set">open set</a>.</li>

<li>In the field of <a href="Convex_analysis" title="Convex analysis">convex analysis</a>, the <a href="Characteristic_function_(convex_analysis)" title="Characteristic function (convex analysis)">characteristic function</a> of a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./826569be03f873b81cdc6f12637ef5520c369d21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.822ex; height:2.176ex;" alt="{\displaystyle A\subset X}" loading="lazy"> is defined differently, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}(x)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c45a188e98b174c54b6732c6d2e3a2a1cbe30c23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.32ex; height:2.843ex;" alt="{\displaystyle \chi _{A}(x)=0}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27bcc9b2afb295d4234bc294860cd0c63bcad2ca.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 x\in A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}(x)=\infty }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \chi _{A}(x)=\infty }</annotation>

</semantics>

</math><img src="./c98fe38591f1e12314eeef7090018f7e2edd4586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.481ex; height:2.843ex;" alt="{\displaystyle \chi _{A}(x)=\infty }" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\notin A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8152431575305d6a6145adf9b279891a65923eba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.913ex; height:2.676ex;" alt="{\displaystyle x\notin A}" loading="lazy">. With that definition, the characteristic function of any closed set is lower semicontinuous, and the characteristic function of any open set is upper semicontinuous.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Binary_operations_on_semicontinuous_functions">Binary operations on semicontinuous functions</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c625c6d786e97e6934d67111373813ffb481978a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.753ex; height:3.343ex;" alt="{\displaystyle f,g:X\to {\overline {\mathbb {R} }}}" loading="lazy">.

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> and <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="./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"> are lower semicontinuous, then the sum <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="./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"> is lower semicontinuous<a href="#cite_note-6">[6]</a> (provided the sum is well-defined, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)+g(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./db982e2fa40cfdcef300a6d0ff9a516be29f7834.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.513ex; height:2.843ex;" alt="{\displaystyle f(x)+g(x)}" loading="lazy"> is not the <a href="Indeterminate_form" title="Indeterminate form">indeterminate form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty +\infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3cb652063a40d7e83d3c5f5f1ca6e74b076837b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.296ex; height:2.176ex;" alt="{\displaystyle -\infty +\infty }" loading="lazy">). The same holds for upper semicontinuous functions.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> and <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="./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"> are lower semicontinuous and non-negative, then the product function <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="./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"> is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.</li>

<li>The function <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="./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"> is lower semicontinuous if and only if <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="./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"> is upper semicontinuous.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> and <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="./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"> are upper semicontinuous and <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="./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"> is <a href="Monotonic_function" title="Monotonic function">non-decreasing</a>, then the <a href="Function_composition" title="Function composition">composition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b2f61ca7838709fbae07dce9c0d513770f10cfae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" loading="lazy"> is upper semicontinuous. On the other hand, if <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="./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"> is not non-decreasing, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b2f61ca7838709fbae07dce9c0d513770f10cfae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" loading="lazy"> may not be upper semicontinuous. For example take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"> defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=-x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c9a57f7495c03ae2b21e25aa228e60247b3f8f54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.654ex; height:2.843ex;" alt="{\displaystyle f(x)=-x}" loading="lazy">. Then <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="./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"> is continuous and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=-g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./592b77ccbc65504c4b3ef29e96e7b6ad9792c583.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.612ex; height:2.509ex;" alt="{\displaystyle f\circ g=-g}" loading="lazy">, which is not upper semicontinuous unless <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="./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"> is continuous.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> and <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="./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"> are lower semicontinuous, their (pointwise) maximum and minimum (defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \max\{f(x),g(x)\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto \max\{f(x),g(x)\}}</annotation>

</semantics>

</math><img src="./c1cdad11696f9badbfa2aad2d5099326585c6587.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.301ex; height:2.843ex;" alt="{\displaystyle x\mapsto \max\{f(x),g(x)\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto \min\{f(x),g(x)\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto \min\{f(x),g(x)\}}</annotation>

</semantics>

</math><img src="./901d42cf726042afdeda57e8b7832c5543e3db4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.85ex; height:2.843ex;" alt="{\displaystyle x\mapsto \min\{f(x),g(x)\}}" loading="lazy">) are also lower semicontinuous. Consequently, the set of all lower semicontinuous functions from <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="./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"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c62ecdf73d36e9d05a018732fd5f51db73d0c55b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.793ex; height:3.009ex;" alt="{\displaystyle {\overline {\mathbb {R} }}}" loading="lazy"> (or to <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="./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">) forms a <a href="Lattice_(order)" title="Lattice (order)">lattice</a>. The corresponding statements also hold for upper semicontinuous functions.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Optimization_of_semicontinuous_functions">Optimization of semicontinuous functions</h3></div>

<ul><li>The (pointwise) <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a> of an arbitrary family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{i})_{i\in I}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f_{i})_{i\in I}}</annotation>

</semantics>

</math><img src="./5f3dd9718b5e3460904d390a0d092237aec266ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.473ex; height:2.843ex;" alt="{\displaystyle (f_{i})_{i\in I}}" loading="lazy"> of lower semicontinuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}:X\to {\overline {\mathbb {R} }}}">

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e713ed81f89c7fdc8c5307fc5478502ca9778228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.263ex; height:3.343ex;" alt="{\displaystyle f_{i}:X\to {\overline {\mathbb {R} }}}" loading="lazy"> (defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sup\{f_{i}(x):i\in I\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7b8f5c859b4908c2c3b07e06cbd85b86b241ffc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.172ex; height:2.843ex;" alt="{\displaystyle f(x)=\sup\{f_{i}(x):i\in I\}}" loading="lazy">) is lower semicontinuous.<a href="#cite_note-7">[7]</a></li></ul>

<dl><dd>In particular, the limit of a <a href="Monotone_increasing" class="mw-redirect" title="Monotone increasing">monotone increasing</a> sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}\leq f_{2}\leq f_{3}\leq \cdots }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}\leq f_{2}\leq f_{3}\leq \cdots }</annotation>

</semantics>

</math><img src="./417cd884a4378159daa240785325830debb40e10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.599ex; height:2.509ex;" alt="{\displaystyle f_{1}\leq f_{2}\leq f_{3}\leq \cdots }" loading="lazy"> of continuous functions is lower semicontinuous. (The Theorem of Baire below provides a partial converse.) The limit function will only be lower semicontinuous in general, not continuous. An example is given by the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}(x)=1-(1-x)^{n}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b14ad2f00a23ac028915cc4a612b85500051cebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.958ex; height:2.843ex;" alt="{\displaystyle f_{n}(x)=1-(1-x)^{n}}" loading="lazy"> defined for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in [0,1]}</annotation>

</semantics>

</math><img src="./64a15936df283add394ab909aa7a5e24e7fb6bb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.823ex; height:2.843ex;" alt="{\displaystyle x\in [0,1]}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,2,\ldots .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1,2,\ldots .}</annotation>

</semantics>

</math><img src="./7232c7585f99f91de353d07f3562bb372b94393b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.643ex; height:2.509ex;" alt="{\displaystyle n=1,2,\ldots .}" loading="lazy"></dd></dl>

<dl><dd>Likewise, the <a href="Infimum" class="mw-redirect" title="Infimum">infimum</a> of an arbitrary family of upper semicontinuous functions is upper semicontinuous. And the limit of a <a href="Monotone_decreasing" class="mw-redirect" title="Monotone decreasing">monotone decreasing</a> sequence of continuous functions is upper semicontinuous.</dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> is a <a href="Compact_space" title="Compact space">compact space</a> (for instance a closed bounded interval <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="./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">) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:C\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./807c3009eb208033c4fd3cccab84f417c91db4cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.389ex; height:3.343ex;" alt="{\displaystyle f:C\to {\overline {\mathbb {R} }}}" loading="lazy"> is upper semicontinuous, then <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="./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"> attains a maximum on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./067be67e68f60c53ce83241748d0d6249675c58d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}" loading="lazy"> If <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="./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"> is lower semicontinuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./64528f031cdbe1f52bdaf4ba7a8401108c0d2dc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle C,}" loading="lazy"> it attains a minimum on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./067be67e68f60c53ce83241748d0d6249675c58d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.413ex; height:2.176ex;" alt="{\displaystyle C.}" loading="lazy"></li></ul>

<dl><dd>(Proof for the upper semicontinuous case: By condition (5) in the definition, <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="./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"> is continuous when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c62ecdf73d36e9d05a018732fd5f51db73d0c55b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.793ex; height:3.009ex;" alt="{\displaystyle {\overline {\mathbb {R} }}}" loading="lazy"> is given the left order topology. So its image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(C)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6b2b925cba46183c6b54011f186728ce596e92ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.854ex; height:2.843ex;" alt="{\displaystyle f(C)}" loading="lazy"> is compact in that topology. And the compact sets in that topology are exactly the sets with a maximum. For an alternative proof, see the article on the <a href="Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a>.)</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Other_properties">Other properties</h3></div>

<ul><li>(Theorem of Baire)<a href="#cite_note-8">[note 1]</a> Let <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="./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"> be a <a href="Metric_space" title="Metric space">metric space</a>. Every lower semicontinuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./809ed19a0ef844995d674e77acd2f0c8b03a3d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.603ex; height:3.343ex;" alt="{\displaystyle f:X\to {\overline {\mathbb {R} }}}" loading="lazy"> is the limit of a point-wise <a href="Monotone_increasing" class="mw-redirect" title="Monotone increasing">increasing</a> sequence of extended real-valued continuous functions on <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="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"> In particular, there exists a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f_{i}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a98b80e62d2a39bb2bdf0dc2c882a3fd4b810c48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.264ex; height:2.843ex;" alt="{\displaystyle \{f_{i}\}}" loading="lazy"> of continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}:X\to {\overline {\mathbb {R} }}}">

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e713ed81f89c7fdc8c5307fc5478502ca9778228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.263ex; height:3.343ex;" alt="{\displaystyle f_{i}:X\to {\overline {\mathbb {R} }}}" loading="lazy"> such that</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}(x)\leq f_{i+1}(x)\quad \forall x\in X,\ \forall i=0,1,2,\dots }</annotation>

</semantics>

</math> and</dd>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{i\to \infty }f_{i}(x)=f(x)\quad \forall x\in X.}</annotation>

</semantics>

</math></dd>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> does not take the value <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="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.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">, the continuous functions can be taken to be real-valued.<a href="#cite_note-9">[8]</a><a href="#cite_note-10">[9]</a></dd></dl>

<dl><dd>Additionally, every upper semicontinuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to {\overline {\mathbb {R} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./809ed19a0ef844995d674e77acd2f0c8b03a3d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.603ex; height:3.343ex;" alt="{\displaystyle f:X\to {\overline {\mathbb {R} }}}" loading="lazy"> is the limit of a <a href="Monotone_decreasing" class="mw-redirect" title="Monotone decreasing">monotone decreasing</a> sequence of extended real-valued continuous functions on <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="./382fc9d9db960b232f5960d73b4a4762c7a047e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X;}" loading="lazy"> if <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="./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"> does not take the value <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="./bf536449e5e9698b2c4d8e1c19804c0f49948223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.971ex; height:2.009ex;" alt="{\displaystyle \infty ,}" loading="lazy"> the continuous functions can be taken to be real-valued.</dd></dl>

<ul><li>Any upper semicontinuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1920b3d1342e353b69a46b6b7bf56a8efbf4ecaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {N} }" loading="lazy"> on an arbitrary topological space <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="./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"> is locally constant on some <a href="Dense_set" title="Dense set">dense open subset</a> of <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="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy"></li></ul>

<ul><li>If the topological space <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="./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"> is <a href="Sequential_space" title="Sequential space">sequential</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./669fa4832da4b0b229d77eadb270e95188f2eb10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f:X\to \mathbb {R} }" loading="lazy"> is upper semi-continuous if and only if it is sequentially upper semi-continuous, that is, if for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> and any sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})_{n}\subset X}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./17ff44b747295cdabb6b56b8993da584fe25bba3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.654ex; height:2.843ex;" alt="{\displaystyle (x_{n})_{n}\subset X}" loading="lazy"> that converges towards <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="./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">, there holds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \limsup _{n\to \infty }f(x_{n})\leqslant f(x)}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./65f9285289fd3184398a17e3c5f8925b403f26fb.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})\leqslant f(x)}" loading="lazy">. Equivalently, in a sequential space, <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="./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"> is upper semicontinuous if and only if its superlevel sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\,x\in X\,|\,f(x)\geqslant y\,\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\,x\in X\,|\,f(x)\geqslant y\,\}}</annotation>

</semantics>

</math><img src="./af91bedb879b4a9c8cb2c89182d49f6e1aee52d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.342ex; height:2.843ex;" alt="{\displaystyle \{\,x\in X\,|\,f(x)\geqslant y\,\}}" loading="lazy"> are <a href="Fr%C3%A9chet%E2%80%93Urysohn_space#Sequentially_open/closed_sets" title="Fréchet–Urysohn space">sequentially closed</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e17075a7abdf541b17004d8b9d0dac081860d685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.674ex; height:2.509ex;" alt="{\displaystyle y\in \mathbb {R} }" loading="lazy">. In general, upper semicontinuous functions are sequentially upper semicontinuous, but the converse may be false.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Semicontinuity_of_set-valued_functions">Semicontinuity of set-valued functions</h2></div>

For <a href="Set-valued_function" title="Set-valued function">set-valued functions</a>, several concepts of semicontinuity have been defined, namely upper, lower, outer, and inner semicontinuity, as well as upper and lower <a href="Hemicontinuity" title="Hemicontinuity">hemicontinuity</a>.

A set-valued function <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="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> from a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> to a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> is written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:A\rightrightarrows B.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:A\rightrightarrows B.}</annotation>

</semantics>

</math><img src="./f25362637d7922f05ba1a9946ef4a7065ae39604.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.446ex; height:2.343ex;" alt="{\displaystyle F:A\rightrightarrows B.}" loading="lazy"> For each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in A,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7c649021f8a56d0234bfa3eae952bcab3edad556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.56ex; height:2.509ex;" alt="{\displaystyle x\in A,}" loading="lazy"> the function <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="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> defines a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)\subset B.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x)\subset B.}</annotation>

</semantics>

</math><img src="./cab0f20fbd85556265f534c8946e546d2ef2ccd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.389ex; height:2.843ex;" alt="{\displaystyle F(x)\subset B.}" loading="lazy">

The <a href="Preimage" class="mw-redirect" title="Preimage">preimage</a> of a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subset B}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b5aa817f046444800dcbf73e490c903a6bb60e7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.362ex; height:2.176ex;" alt="{\displaystyle S\subset B}" loading="lazy"> under <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="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is defined as

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{-1}(S):=\{x\in A:F(x)\cap S\neq \varnothing \}.}</annotation>

</semantics>

</math>

That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{-1}(S)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bede3081f80f84b43383a55546ec90ac7d3944dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.456ex; height:3.176ex;" alt="{\displaystyle F^{-1}(S)}" loading="lazy"> is the set that contains every point <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="./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 A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"> is not <a href="Disjoint_sets" title="Disjoint sets">disjoint</a> from <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="./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">.<a href="#cite_note-freemanRobust-11">[10]</a>

<div class="mw-heading mw-heading3"><h3 id="Upper_and_lower_semicontinuity">Upper and lower semicontinuity</h3></div>

A set-valued map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./577ddd72ce65d4a7851c35a549aac59affd2d337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.542ex; height:2.509ex;" alt="{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}" loading="lazy"> is upper semicontinuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./648f0dff0504d57613fa1d129393bb96013d909c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.523ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{m}}" loading="lazy"> if for every open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c1caefb347c86337ea7cd0c354acd2294bd7d81d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.778ex; height:2.343ex;" alt="{\displaystyle U\subset \mathbb {R} ^{n}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)\subset U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5a46730bf12b29a4d5238cc645bcbf30ff33b35e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.761ex; height:2.843ex;" alt="{\displaystyle F(x)\subset U}" loading="lazy">, there exists a neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> of <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="./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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(V)\subset U.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(V)\subset U.}</annotation>

</semantics>

</math><img src="./dc39515464c2ede290d121973cd678f2b18eff24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.865ex; height:2.843ex;" alt="{\displaystyle F(V)\subset U.}" loading="lazy"><a href="#cite_note-freemanRobust-11">[10]</a>: Def. 2.1

A set-valued map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./577ddd72ce65d4a7851c35a549aac59affd2d337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.542ex; height:2.509ex;" alt="{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}" loading="lazy"> is lower semicontinuous at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./648f0dff0504d57613fa1d129393bb96013d909c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.523ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{m}}" loading="lazy"> if for every open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c1caefb347c86337ea7cd0c354acd2294bd7d81d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.778ex; height:2.343ex;" alt="{\displaystyle U\subset \mathbb {R} ^{n}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in F^{-1}(U),}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in F^{-1}(U),}</annotation>

</semantics>

</math><img src="./758b8e9235e17140cbfabca60d208ad0aca30914.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.557ex; height:3.176ex;" alt="{\displaystyle x\in F^{-1}(U),}" loading="lazy"> there exists a neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./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"> of <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="./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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\subset F^{-1}(U).}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\subset F^{-1}(U).}</annotation>

</semantics>

</math><img src="./84c62abcf3b1658cfe5b46bcef751c6999e00bec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.272ex; height:3.176ex;" alt="{\displaystyle V\subset F^{-1}(U).}" loading="lazy"><a href="#cite_note-freemanRobust-11">[10]</a>: Def. 2.2

Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6a87a024931038d1858dc22e8a194e5978c3412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> in the above definitions with arbitrary topological spaces.<a href="#cite_note-freemanRobust-11">[10]</a>

Note, that there is not a direct correspondence between single-valued lower and upper semicontinuity and set-valued lower and upper semicontinuouty.

An upper semicontinuous single-valued function is not necessarily upper semicontinuous when considered as a set-valued map.<a href="#cite_note-freemanRobust-11">[10]</a>: 18

For example, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1e3a10a3ad05781f5cf9c2d875a02227e21a8448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} }" loading="lazy"> defined by

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}-1&{\mbox{if }}x<0,\\1&{\mbox{if }}x\geq 0\end{cases}}}</annotation>

</semantics>

</math>

is upper semicontinuous in the single-valued sense but the set-valued map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto F(x):=\{f(x)\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5f981d39bb3f76e8abcae155c54d005b0231e36f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.311ex; height:2.843ex;" alt="{\displaystyle x\mapsto F(x):=\{f(x)\}}" loading="lazy"> is not upper semicontinuous in the set-valued sense.

<div class="mw-heading mw-heading3"><h3 id="Inner_and_outer_semicontinuity">Inner and outer semicontinuity</h3></div>

A set-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./577ddd72ce65d4a7851c35a549aac59affd2d337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.542ex; height:2.509ex;" alt="{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}" loading="lazy"> is called inner semicontinuous at <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="./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"> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in F(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cc1d00c8a170334da8953bca479543c60caf1d61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.876ex; height:2.843ex;" alt="{\displaystyle y\in F(x)}" loading="lazy"> and every convergent sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d65f04ce52a62128b2bcc7f11353b4e77374658d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.939ex; height:2.843ex;" alt="{\displaystyle (x_{i})}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6a87a024931038d1858dc22e8a194e5978c3412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\to x}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d0d9c85886cf5e76ba17a4125f71bb0e3718bbaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.073ex; height:2.176ex;" alt="{\displaystyle x_{i}\to x}" loading="lazy">, there exists

a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c11054157cb27ceec5584a202820ab82dd2d5791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.748ex; height:2.843ex;" alt="{\displaystyle (y_{i})}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}\to y}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b58d6da412357422314ba7911998a3a7921bed8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.708ex; height:2.176ex;" alt="{\displaystyle y_{i}\to y}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}\in F\left(x_{i}\right)}">

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}\in F\left(x_{i}\right)}</annotation>

</semantics>

</math><img src="./5a4ebf257e528813caf8790b01c467dd9003e831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.846ex; height:2.843ex;" alt="{\displaystyle y_{i}\in F\left(x_{i}\right)}" loading="lazy"> for all sufficiently large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \mathbb {N} .}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cbd85387f5ee7c9efbaced07edbe9feb9af82d50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.968ex; height:2.176ex;" alt="{\displaystyle i\in \mathbb {N} .}" loading="lazy"><a href="#cite_note-goebelSetvalued-12">[11]</a><a href="#cite_note-13">[note 2]</a>

A set-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./577ddd72ce65d4a7851c35a549aac59affd2d337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.542ex; height:2.509ex;" alt="{\displaystyle F:\mathbb {R} ^{m}\rightrightarrows \mathbb {R} ^{n}}" loading="lazy"> is called outer semicontinuous at <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="./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"> if for every convergence sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d65f04ce52a62128b2bcc7f11353b4e77374658d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.939ex; height:2.843ex;" alt="{\displaystyle (x_{i})}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6a87a024931038d1858dc22e8a194e5978c3412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\to x}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d0d9c85886cf5e76ba17a4125f71bb0e3718bbaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.073ex; height:2.176ex;" alt="{\displaystyle x_{i}\to x}" loading="lazy"> and every convergent sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c11054157cb27ceec5584a202820ab82dd2d5791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.748ex; height:2.843ex;" alt="{\displaystyle (y_{i})}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}\in F(x_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}\in F(x_{i})}</annotation>

</semantics>

</math><img src="./66af5b947482936b99e8eaa7e50e62056e235cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.459ex; height:2.843ex;" alt="{\displaystyle y_{i}\in F(x_{i})}" loading="lazy"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \mathbb {N} ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2ebc93b067b01dc98b77de4a3411ccf9f93bf181.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.968ex; height:2.509ex;" alt="{\displaystyle i\in \mathbb {N} ,}" loading="lazy"> the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c11054157cb27ceec5584a202820ab82dd2d5791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.748ex; height:2.843ex;" alt="{\displaystyle (y_{i})}" loading="lazy"> converges to a point in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./71a82805d469cdfa7856c11d6ee756acd1dc7174.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}" loading="lazy"> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{i\to \infty }y_{i}\in F(x)}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{i\to \infty }y_{i}\in F(x)}</annotation>

</semantics>

</math><img src="./de654391bbfbc89ba4e96ee341ecee37846f7a9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.9ex; height:4.009ex;" alt="{\displaystyle \lim _{i\to \infty }y_{i}\in F(x)}" loading="lazy">).<a href="#cite_note-goebelSetvalued-12">[11]</a>

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

Because the supremum of a family of lower semicontinuous functions is lower semicontinuous, if <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="./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"> is an arbitrary extended-real valued function on a topological space <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="./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">, the supremum of the set of lower semicontinuous functions majorized by <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="./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"> is lower semicontinuous. This greatest lower semicontinuous function majorized by <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="./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"> is the lower semicontinuous hull of <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="./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">.<a href="#cite_note-FOOTNOTERockafellar1970-14">[12]</a> The hull <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8d9e0161729e04d187152ed5e25cca64312eed39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.068ex; height:2.843ex;" alt="{\displaystyle H_{f}}" loading="lazy"> is defined pointwise by the relation<a href="#cite_note-FOOTNOTEBourbaki1966IV.6.2-15">[13]</a>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mstyle>

</mrow>

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

</semantics>

</math>

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

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8d9e0161729e04d187152ed5e25cca64312eed39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.068ex; height:2.843ex;" alt="{\displaystyle H_{f}}" loading="lazy"> has the property that its epigraph is the closure of the epigraph of <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="./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">.

The lower semicontinuous hull plays a role in <a href="Convex_analysis" title="Convex analysis">convex analysis</a>. Given a convex (extended real) function, the epigraph might not be closed. But the lower semicontinuous hull of a convex function is convex, and is known as the closure of the original convex function.

Some operations in convex analysis, such as the <a href="Legendre_transform" class="mw-redirect" title="Legendre transform">Legendre transform</a> automatically produce closed convex functions. The Legendre transform applied twice to a convex function gives the closure of the original function, rather than the original function. Thus the lower semicontinuous hull is a way of regularizing convex functions, by modifying it at boundary points of its effective domain.

In <a href="Category_theory" title="Category theory">categorical terms</a>, the lower semicontinuous hull of a function <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="./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"> is the (left) <a href="Kan_extension" title="Kan extension">Kan extension</a> of <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="./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"> along the inclusion of the poset of open neighborhoods (ordered by reverse inclusion) into the topological space <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="./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">. Explicitly, the value of the hull <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{f}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8d9e0161729e04d187152ed5e25cca64312eed39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.068ex; height:2.843ex;" alt="{\displaystyle H_{f}}" loading="lazy"> at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> is given by the colimit:

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathrm {Lan} _{\iota }f)(x)=\inf _{U\ni x}\sup _{y\in U}f(y),}</annotation>

</semantics>

</math>

which coincides with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \liminf _{y\to x}f(y)}">

</mrow>

</munder>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./32f5d961a1c9c7fcb6db763f303816ed45c92d4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.899ex; height:4.176ex;" alt="{\displaystyle \liminf _{y\to x}f(y)}" loading="lazy">, the left Kan extension under the inclusion functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bce48dd56254d0a7c33e987c7c8eeb44c963ac04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.823ex; height:1.676ex;" alt="{\displaystyle \iota }" loading="lazy">. In this formulation, the process of taking the semicontinuous envelope is a special case of the Kan extension machinery in enriched category theory. The upper semicontinuous hull is a right Kan extension.

Other types of hulls are often considered in applications. For example, the infimum of the set of continuous <a href="Affine_function" class="mw-redirect" title="Affine function">affine functions</a> that majorize a given function on a convex subset of a topological vector space is upper semicontinuous. This fact is used in the proof of the <a href="Choquet_theorem" class="mw-redirect" title="Choquet theorem">Choquet theorem</a>.<a href="#cite_note-FOOTNOTEPhelps1966Chapter_3-16">[14]</a> Similar ideas applied to <a href="Subharmonic_function" title="Subharmonic function">subharmonic functions</a> are used in the <a href="Perron_method" title="Perron method">Perron method</a> for solving the <a href="Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a> for the <a href="Laplace_operator" title="Laplace operator">Laplace operator</a> in a domain. The key condition for the class of subharmonic solutions is upper semicontinuity, particularly near the boundary where the boundary conditions are applied.

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

<div class="mw-heading mw-heading3"><h3 id="Calculus_of_variations">Calculus of variations</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Direct_methods_in_the_calculus_of_variations" class="mw-redirect" title="Direct methods in the calculus of variations">Direct methods in the calculus of variations</a></div>

An important application of semicontinuity is to the <a href="Calculus_of_variations" title="Calculus of variations">calculus of variations</a>. It derives its significance in this context due to the following theorem.<a href="#cite_note-FOOTNOTEGiusti2003-17">[15]</a> Let <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="./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"> be a topological space, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:X\to (-\infty ,+\infty ]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:X\to (-\infty ,+\infty ]}</annotation>

</semantics>

</math><img src="./d1d24deeeed09dea31c47ec03dd29a73842d5ebb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.121ex; height:2.843ex;" alt="{\displaystyle F:X\to (-\infty ,+\infty ]}" loading="lazy">. A minimizing sequence is a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" 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="./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"> such that

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{k\to \infty }F(x_{k})=\inf _{X}F.}</annotation>

</semantics>

</math>

The theorem is that if <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="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is sequentially lower semicontinuous and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy"> is a minimizing sequence that converges to <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="./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">, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{0})=\inf _{X}F.}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{0})=\inf _{X}F.}</annotation>

</semantics>

</math>

That is, <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="./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"> is an absolute minimum of <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="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">.

This is often combined with results such as <a href="Tonelli's_theorem_(functional_analysis)" title="Tonelli's theorem (functional analysis)">Tonelli's theorem</a> in <a href="Functional_analysis" title="Functional analysis">functional analysis</a>, which characterizes the <a href="Weak_topology" title="Weak topology">weak</a> lower semicontinuity of <a href="Nonlinearity" class="mw-redirect" title="Nonlinearity">nonlinear</a> <a href="Functional_(mathematics)" title="Functional (mathematics)">functionals</a> on <a href="Lp_space" title="Lp space">Lp spaces</a> in terms of the convexity of another function. More specialized results of this kind are useful in variational formulations of problems in <a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>, which relate semicontinuity of functionals given by integration to the convexity properties of the integrand, often defined on some <a href="Sobolev_space" title="Sobolev space">Sobolev space</a>. The prototypical example is the <a href="Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a> for the <a href="Laplace_operator" title="Laplace operator">Laplace operator</a>, which can be formulated as a minimization problem of the <a href="Energy" title="Energy">energy</a>, subject to <a href="Boundary_conditions" class="mw-redirect" title="Boundary conditions">boundary conditions</a>,

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(u)=\int _{\Omega }|\nabla u|^{2},}</annotation>

</semantics>

</math>

i.e., the integral of the squared norm of the gradient of a function over a bounded domain in Euclidean space. The integrand is convex in an appropriate Sobolev space, so the limit of a minimizing sequence is a solution of the Dirichlet problem. This has implications, for instance, for <a href="Finite_element" class="mw-redirect" title="Finite element">finite element</a> solutions, which gives a way to construct a minimizing sequence.

<div class="mw-heading mw-heading3"><h3 id="Existence_of_saddle_points">Existence of saddle points</h3></div>

Together with convexity assumptions, both upper and lower semicontinuity play a role in theorems guaranteeing the existence of <a href="Saddle_point" title="Saddle point">saddle points</a> of functions, on <a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">locally convex topological vector spaces</a>. One such result is the <a href="Minimax_theorem#Sion's_minimax_theorem" title="Minimax theorem">minimax theorem of Fan and Sion</a>.<a href="#cite_note-FOOTNOTEEkelandTémam1999Chapter_4-18">[16]</a> It states that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\times Y\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e2c7bc3330b76d58cdb3bff67e7b0ec60a2509c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.102ex; height:2.509ex;" alt="{\displaystyle f:X\times Y\to \mathbb {R} }" loading="lazy"> is a function from a pair of non-empty closed, convex sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8705438171d938b7f59cd1bfa5b7d99b6afa5cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}" loading="lazy"> belonging to <a href="Reflexive_Banach_space" class="mw-redirect" title="Reflexive Banach space">reflexive Banach spaces</a>, such that

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a935de383489e2da409e85c2d5f7921335bd3acd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.098ex; height:2.843ex;" alt="{\displaystyle f(x,\cdot )}" loading="lazy"> is concave and upper semicontinuous for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> and</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0228644022235d794440bf2762ec97a8f074e47b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.924ex; height:2.843ex;" alt="{\displaystyle f(\cdot ,y)}" loading="lazy"> is convex and lower semicontinuous for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy">,</li></ul>

then the set of saddle points of <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="./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"> is convex. If both convexity and concavity are strict, then there is at most one saddle point. If the sets <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="./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"> and <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="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are bounded, then the set of saddle points is non-empty. A saddle point is by definition a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./29c296094af9a1c665425debeac5eaab99a37a04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}" loading="lazy"> at which

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</munder>

</mrow>

</munder>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{0},y_{0})=\inf _{x\in X}\sup _{y\in Y}f(x,y)=\sup _{y\in Y}\inf _{x\in X}f(x,y).}</annotation>

</semantics>

</math>

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

Many integer-valued functions of importance are also semicontinuous. For a simple example, suppose one has a <a href="Polyhedron" title="Polyhedron">polyhedron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> (or, more generally, a closed convex set) in an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensional vector space. A face of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is by definition the set of maxima of some linear functional on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Define the function

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\inf\{\dim F|F{\text{ is a face of }}K{\text{ and }}x\in F\}.}</annotation>

</semantics>

</math>

Then <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="./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"> is lower semicontinuous. This is intuitively because under any small perturbations, you can move from a face of lower dimension, such as an edge or vertex, to one of higher dimension, but any point of a higher dimensional face cannot be moved to one of lower dimension if the perturbation is small enough.

Another example of a similar character is that <a href="Matrix_rank" class="mw-redirect" title="Matrix rank">matrix rank</a> is a lower semicontinuous function on the space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d82325a2a02ad79bc7c347ba9702ad46eb0de824.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle n\times m}" loading="lazy"> matrices. This is because the rank can go up at matrices which are nearby, but not down. As a result of this, together with the <a href="Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a>, when a <a href="Lie_group" title="Lie group">Lie group</a> acts smoothly on a <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>, the dimension of the orbit through a point is lower semicontinuous (i.e., the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\dim(G\cdot x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1c0704decb35b26a47c04139de6174cca0465cb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.036ex; height:2.843ex;" alt="{\displaystyle f(x)=\dim(G\cdot x)}" loading="lazy">).<a href="#cite_note-19">[17]</a>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_geometry">Algebraic geometry</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Semicontinuity_theorem" class="mw-redirect" title="Semicontinuity theorem">Semicontinuity theorem</a></div>

More sophisticated versions of this same idea play a fundamental role in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, where many dimension maps with codomain in the integers are known to be semicontinuous. (For example as applied to a <a href="Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</a>.)

In general, let <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="./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"> and <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="./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"> be <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a21a941b35354b73f5a62849067deac7ee9794dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.309ex; height:2.509ex;" alt="{\displaystyle f:X\to S}" loading="lazy"> a flat and proper morphism of finite presentation. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> be an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy">-module flat and of finite presentation over <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="./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">. Then for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0ae4e6baa4f5f76345756ffed7375ad64b0f0ecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.193ex; height:2.176ex;" alt="{\displaystyle i\in \mathbb {Z} }" loading="lazy"> the function

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{i}:S\to \mathbb {Z} _{\geq 0},s\mapsto {\text{dim}}_{\kappa (s)}H^{i}(X_{s},{\mathcal {F}}_{s})}</annotation>

</semantics>

</math>

is upper semicontinuous.<a href="#cite_note-20">[18]</a> An important special case of this theorem when additionally <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e271362eae51b4e4190a78759a6ba0f524c622f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.513ex; height:2.509ex;" alt="{\displaystyle X,S}" loading="lazy"> are noetherian, <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="./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"> is projective and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> is coherent can be found in the standard textbook of <a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne</a>.<a href="#cite_note-AG-21">[19]</a>: 288 Original work in the language of hypercohomology can be found in <a href="%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique" title="Éléments de géométrie algébrique">EGA III</a><a href="#cite_note-22">[20]</a> Théorème (7.7.5), citing also previous work, in particular <a href="Hans_Grauert" title="Hans Grauert">Grauert</a> for the complex-analytic setting.

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8705438171d938b7f59cd1bfa5b7d99b6afa5cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.787ex; height:2.509ex;" alt="{\displaystyle X,Y}" loading="lazy"> be schemes and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> a morphism of finite type. The function

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{X/Y}:Y\to \mathbb {Z} _{\geq 0}\cup \{\infty \},y\mapsto {\text{dim}}_{\text{top}}X_{y}}</annotation>

</semantics>

</math>

associates to any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cee1c0ec36a82f33f5e3d7434d5667881b4ec323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.769ex; height:2.509ex;" alt="{\displaystyle y\in Y}" loading="lazy"> the dimension of the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./00ba221e3f5c92da8c1bd62dcad2083e7493160d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.974ex; height:2.843ex;" alt="{\displaystyle X_{y}}" loading="lazy">. If <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="./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"> is a flat morphism of schemes of finite presentation, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{X/Y}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3adb0564576eb4a4b776d302b6b67172c104f6e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.103ex; height:2.509ex;" alt="{\displaystyle n_{X/Y}}" loading="lazy"> is lower semicontinuous.<a href="#cite_note-23">[21]</a> If <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="./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"> is a proper morphism of schemes, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{X/Y}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3adb0564576eb4a4b776d302b6b67172c104f6e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.103ex; height:2.509ex;" alt="{\displaystyle n_{X/Y}}" loading="lazy"> is upper semicontinuous.<a href="#cite_note-24">[22]</a>

<a href="Ravi_Vakil" title="Ravi Vakil">Vakil</a> collected a list of further semicontinuity results in algebraic geometry.<a href="#cite_note-25">[23]</a>

<div class="mw-heading mw-heading3"><h3 id="Descriptive_set_theory">Descriptive set theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Wadge_hierarchy" title="Wadge hierarchy">Wadge hierarchy</a></div>

Semicontinuous functions are used in <a href="Descriptive_set_theory" title="Descriptive set theory">descriptive set theory</a> to define stratifications of topological spaces by complexity measures such as dimension, rank, or ordinal height.<a href="#cite_note-26">[24]</a><a href="#cite_note-27">[25]</a><a href="#cite_note-28">[26]</a> Such functions often take values in an <a href="Ordinal_number" title="Ordinal number">ordinal</a>, and their semicontinuity ensures that the sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x:f(x)\geq \alpha \}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dbe4983c5a4737de4dc2cbdcda161a72476cb22d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.595ex; height:2.843ex;" alt="{\displaystyle \{x:f(x)\geq \alpha \}}" loading="lazy"> are closed (and hence <a href="Borel_set" title="Borel set">Borel</a> in a <a href="Polish_space" title="Polish space">Polish space</a>).

A central example is the rank function on well-founded trees. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}\subseteq \omega ^{<\omega }}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}\subseteq \omega ^{<\omega }}</annotation>

</semantics>

</math><img src="./d1b1f2201aaee03640334aef6671eb0bfc931dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.013ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}}\subseteq \omega ^{<\omega }}" loading="lazy"> be a tree coded by a point in <a href="Baire_space" title="Baire space">Baire space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{\omega }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9aa5636cabcea62c44c8b91fd9095e06054a6fa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.7ex; height:2.343ex;" alt="{\displaystyle \omega ^{\omega }}" loading="lazy">. The rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ({\mathcal {T}})\in \omega _{1}\cup \{\infty \}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ({\mathcal {T}})\in \omega _{1}\cup \{\infty \}}</annotation>

</semantics>

</math><img src="./3e0a4264e06bbb49976d942551c6e118cd128aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.519ex; height:2.843ex;" alt="{\displaystyle \rho ({\mathcal {T}})\in \omega _{1}\cup \{\infty \}}" loading="lazy"> is defined as the supremum of the lengths of descending sequences in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8236d074e42310f5dc24d1d2b5b8f5981c3e87ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}" loading="lazy">. The function assigning the rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ({\mathcal {T}})}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ({\mathcal {T}})}</annotation>

</semantics>

</math><img src="./4289c508ecab0071ea4b9c1a304b38e85d560e31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \rho ({\mathcal {T}})}" loading="lazy"> to each tree is lower semicontinuous with respect to the natural topology on tree codes. This rank stratifies the space of trees into closed sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\mathcal {T}}:\rho ({\mathcal {T}})\geq \alpha \}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\mathcal {T}}:\rho ({\mathcal {T}})\geq \alpha \}}</annotation>

</semantics>

</math><img src="./93ec962504c78a435e9988b1fdded567343191e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.731ex; height:2.843ex;" alt="{\displaystyle \{{\mathcal {T}}:\rho ({\mathcal {T}})\geq \alpha \}}" loading="lazy">, analogous to how <a href="Matrix_rank" class="mw-redirect" title="Matrix rank">matrix rank</a> stratifies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n\times m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./94cb9d4a4bc62be08e90334fa87531883301210b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n\times m}}" loading="lazy">.

More generally, ordinal-valued lower semicontinuous functions are used to measure the complexity of points or structures in a <a href="Polish_space" title="Polish space">Polish space</a>—such as Scott ranks of countable structures, projective ranks of sets, or Lusin–Novikov complexities of equivalence relations. These functions enable fine classification and are crucial in defining universal sets and effective parametrizations in higher levels of the <a href="Projective_hierarchy" title="Projective hierarchy">projective hierarchy</a>.

Because the preimage of an interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\alpha ,\infty ]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e4d4c99f5ec3cbeef3dfd0d90a906abb2cf30337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.139ex; height:2.843ex;" alt="{\displaystyle [\alpha ,\infty ]}" loading="lazy"> under a lower semicontinuous function is closed, such functions yield canonical stratifications of topological spaces into closed (thus Borel) pieces of increasing complexity. This property is often used in proofs of <a href="Reflection_principle" title="Reflection principle">reflection principles</a>, <a href="Separation_theorem" title="Separation theorem">separation theorems</a>, and in the effective classification of <a href="Borel_equivalence_relation" title="Borel equivalence relation">Borel equivalence relations</a>.

<div class="mw-heading mw-heading3"><h3 id="Dynamical_systems">Dynamical systems</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Entropy_(information_theory)" title="Entropy (information theory)">Entropy (information theory)</a> and <a href="Invariant_measure" title="Invariant measure">Invariant measure</a></div>

In <a href="Ergodic_theory" title="Ergodic theory">ergodic theory</a> and <a href="Topological_dynamics" title="Topological dynamics">topological dynamics</a>, semicontinuity arises naturally when studying functionals on the space of <a href="Invariant_measure" title="Invariant measure">invariant measures</a> of a <a href="Dynamical_system" title="Dynamical system">dynamical system</a>. The most important example is the entropy function, which assigns to each invariant measure its <a href="Measure-theoretic_entropy" class="mw-redirect" title="Measure-theoretic entropy">measure-theoretic entropy</a>.<a href="#cite_note-29">[27]</a><a href="#cite_note-30">[28]</a><a href="#cite_note-31">[29]</a>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,T)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./627ce34fb2db641fafd1e88cf307a668d1789ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle (X,T)}" loading="lazy"> be a topological dynamical system with <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="./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"> compact and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:X\to X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ff1aa55687228075ec7c550d63a5b54c3930efd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.148ex; height:2.176ex;" alt="{\displaystyle T:X\to X}" loading="lazy"> continuous. The space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{T}(X)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{T}(X)}</annotation>

</semantics>

</math><img src="./29a9006c0b97563e8ccd533f6c47913059dbfed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.969ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{T}(X)}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">-invariant Borel probability measures is a compact convex subset of the dual of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(X)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b3297669ebb5d9d3e5ce8d118b7ba75262c8bca5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.556ex; height:2.843ex;" alt="{\displaystyle C(X)}" loading="lazy"> under the weak-* topology. The entropy map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \mapsto h_{\mu }(T)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu \mapsto h_{\mu }(T)}</annotation>

</semantics>

</math><img src="./36b0aee4311a89b5a84bc30d49b4bb2612b6b3eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.024ex; height:3.009ex;" alt="{\displaystyle \mu \mapsto h_{\mu }(T)}" loading="lazy"> is an upper semicontinuous function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{T}(X)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{T}(X)}</annotation>

</semantics>

</math><img src="./29a9006c0b97563e8ccd533f6c47913059dbfed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.969ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{T}(X)}" loading="lazy">:

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \limsup _{\mu _{n}\to \mu }h_{\mu _{n}}(T)\leq h_{\mu }(T).}</annotation>

</semantics>

</math>

This property plays a key role in the <a href="Variational_principle" title="Variational principle">variational principle</a>, which asserts that the topological entropy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\mathrm {top} }(T)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{\mathrm {top} }(T)}</annotation>

</semantics>

</math><img src="./362c32f5c92d28f7a2432baaf910c064024a530c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.392ex; height:3.009ex;" alt="{\displaystyle h_{\mathrm {top} }(T)}" loading="lazy"> is the supremum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{\mu }(T)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{\mu }(T)}</annotation>

</semantics>

</math><img src="./370a217e5bfa9d00a481036493ecf8fd5a803048.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.008ex; height:3.009ex;" alt="{\displaystyle h_{\mu }(T)}" loading="lazy"> over all invariant measures. Upper semicontinuity guarantees that this supremum is attained when the space of measures is compact.

More generally, many functionals of interest—such as <a href="Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponents</a>, dimension spectra, or <a href="Return_time" class="mw-redirect" title="Return time">return time</a> statistics—are semicontinuous on the space of invariant measures. In some cases, these semicontinuity properties are used to prove existence of measures maximizing or minimizing a given quantity, or to establish structural properties of the <a href="Choquet_theory" title="Choquet theory">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{T}(X)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{T}(X)}</annotation>

</semantics>

</math><img src="./29a9006c0b97563e8ccd533f6c47913059dbfed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.969ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{T}(X)}" loading="lazy"> (e.g., that ergodic measures form a residual—dense <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\delta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7b42b79015590af928aa28bda9514373aeac0e1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.8ex; height:2.509ex;" alt="{\displaystyle G_{\delta }}" loading="lazy">—set).

Similar ideas appear in the theory of joinings, where one studies invariant couplings between systems. The set of joinings is compact in the weak-* topology, and semicontinuity is used to analyze disjointness and uniqueness of invariant couplings.

<ul><li><a href="Left-continuous" class="mw-redirect" title="Left-continuous">Directional continuity</a> – Mathematical function with no sudden changesPages displaying short descriptions of redirect targets</li>

<li><a href="Kat%C4%9Btov%E2%80%93Tong_insertion_theorem" title="Katětov–Tong insertion theorem">Katětov–Tong insertion theorem</a> – On existence of a continuous function between semicontinuous upper and lower bounds</li>

<li><a href="Hemicontinuity" title="Hemicontinuity">Hemicontinuity</a> – Semicontinuity for set-valued functions</li>

<li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg</a> – Right continuous function with left limits</li></ul>

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

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

<li id="cite_note-8"><a href="#cite_ref-8">^</a> The result was proved by René Baire in 1904 for real-valued function defined on <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="./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">. It was extended to metric spaces by <a href="Hans_Hahn_(mathematician)" title="Hans Hahn (mathematician)">Hans Hahn</a> in 1917, and <a href="Hing_Tong" title="Hing Tong">Hing Tong</a> showed in 1952 that the most general class of spaces where the theorem holds is the class of <a href="Perfectly_normal_space" class="mw-redirect" title="Perfectly normal space">perfectly normal spaces</a>. (See Engelking, Exercise 1.7.15(c), p. 62 for details and specific references.)

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> In particular, there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{0}\geq 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./32cc8dc1b17bebd2a0bb7134e04c9ab21cf07e97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.118ex; height:2.509ex;" alt="{\displaystyle i_{0}\geq 0}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}\in F(x_{i})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}\in F(x_{i})}</annotation>

</semantics>

</math><img src="./66af5b947482936b99e8eaa7e50e62056e235cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.459ex; height:2.843ex;" alt="{\displaystyle y_{i}\in F(x_{i})}" loading="lazy"> for every natural number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\geq i_{0},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4d0d15a3fe42ff210b9339fa4b5c6b2ef4f1c13b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.404ex; height:2.509ex;" alt="{\displaystyle i\geq i_{0},}" loading="lazy">. The necessisty of only considering the tail of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"> comes from the fact that for small values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3d0f7dadba3056fa3c06a6bee5c0b4182471152.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle i,}" loading="lazy"> the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x_{i})}</annotation>

</semantics>

</math><img src="./54290e388c0dd2693499b3e01585c9755079c16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.679ex; height:2.843ex;" alt="{\displaystyle F(x_{i})}" loading="lazy"> may be empty.

</li>

</ol></div></div>

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

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFVerry" class="citation web cs1">Verry, Matthieu. <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/351274714">"Histoire des mathématiques - René Baire"</a>.</cite>

</li>

<li id="cite_note-Stromberg-2">^ <a href="#cite_ref-Stromberg_2-0">a</a> <a href="#cite_ref-Stromberg_2-1">b</a> Stromberg, p. 132, Exercise 4

</li>

<li id="cite_note-Kurdila2005-3"><a href="#cite_ref-Kurdila2005_3-0">^</a> <cite id="CITEREFKurdila,_A._J.Zabarankin,_M.2005" class="citation book cs1 cs1-prop-vanc-accept">Kurdila, A. J., Zabarankin, M. (2005). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/3-7643-7357-1_7">"Lower Semicontinuous Functionals"</a>. Convex Functional Analysis. Systems & Control: Foundations & Applications (1st ed.). Birkhäuser-Verlag. pp. 205–219. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-7643-7357-1_7">10.1007/3-7643-7357-1_7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-2198-7</bdi>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Willard, p. 49, problem 7K

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFGiaquinta2007" class="citation book cs1">Giaquinta, Mariano (2007). Mathematical analysis : linear and metric structures and continuity. Giuseppe Modica (1 ed.). Boston: Birkhäuser. Theorem 11.3, p.396. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4514-4</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/213079540">213079540</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFPuterman2005" class="citation book cs1">Puterman, Martin L. (2005). <a rel="nofollow" class="external text" href="https://archive.org/details/markovdecisionpr00pute_298">Markov Decision Processes Discrete Stochastic Dynamic Programming</a>. Wiley-Interscience. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/markovdecisionpr00pute_298/page/n618">602</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-72782-8</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/1662726">"To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous"</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> Stromberg, p. 132, Exercise 4(g)

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/1279763">"Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions"</a>.</cite>

</li>

<li id="cite_note-freemanRobust-11">^ <a href="#cite_ref-freemanRobust_11-0">a</a> <a href="#cite_ref-freemanRobust_11-1">b</a> <a href="#cite_ref-freemanRobust_11-2">c</a> <a href="#cite_ref-freemanRobust_11-3">d</a> <a href="#cite_ref-freemanRobust_11-4">e</a> <cite id="CITEREFFreeman,_R._A.Kokotović,_P.1996" class="citation book cs1 cs1-prop-vanc-accept">Freeman, R. A., Kokotović, P. (1996). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/978-0-8176-4759-9">Robust Nonlinear Control Design</a>. Birkhäuser Boston. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-8176-4759-9">10.1007/978-0-8176-4759-9</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4758-2</bdi>.</cite>.

</li>

<cite id="CITEREFGoebel,_R._K.2024" class="citation book cs1 cs1-prop-vanc-accept">Goebel, R. K. (January 2024). <a rel="nofollow" class="external text" href="https://epubs.siam.org/doi/10.1137/1.9781611977981.ch2">"Chapter 2: Set convergence and set-valued mappings"</a>. Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction. Other Titles in Applied Mathematics. Society for Industrial and Applied Mathematics. pp. 21–36. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611977981.ch2">10.1137/1.9781611977981.ch2</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-61197-797-4</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTERockafellar1970-14"><a href="#cite_ref-FOOTNOTERockafellar1970_14-0">^</a> <a href="#CITEREFRockafellar1970">Rockafellar 1970</a>.

</li>

<li id="cite_note-FOOTNOTEBourbaki1966IV.6.2-15"><a href="#cite_ref-FOOTNOTEBourbaki1966IV.6.2_15-0">^</a> <a href="#CITEREFBourbaki1966">Bourbaki 1966</a>, IV.6.2. sfn error: no target: CITEREFBourbaki1966 (help)

</li>

<li id="cite_note-FOOTNOTEPhelps1966Chapter_3-16"><a href="#cite_ref-FOOTNOTEPhelps1966Chapter_3_16-0">^</a> <a href="#CITEREFPhelps1966">Phelps 1966</a>, Chapter 3.

</li>

<li id="cite_note-FOOTNOTEGiusti2003-17"><a href="#cite_ref-FOOTNOTEGiusti2003_17-0">^</a> <a href="#CITEREFGiusti2003">Giusti 2003</a>.

</li>

<li id="cite_note-FOOTNOTEEkelandTémam1999Chapter_4-18"><a href="#cite_ref-FOOTNOTEEkelandTémam1999Chapter_4_18-0">^</a> <a href="#CITEREFEkelandTémam1999">Ekeland & Témam 1999</a>, Chapter 4.

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> Alexandrino, M. M. and Bettiol, R. G.(2015). Proper and isometric group actions. In Lecture Notes of the Unione Matematica Italiana (Vol. 23, pp. 51-84). Springer.

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/0BDN">"Stacks Project — Lemma 36.32.1"</a>. Stacks.math.columbia.edu. Retrieved 2025-04-27.</cite>

</li>

<li id="cite_note-AG-21"><a href="#cite_ref-AG_21-0">^</a> <cite id="CITEREFHartshorne1977" class="citation cs2"><a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3rtX9t-nnvwC">Algebraic Geometry</a>, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90244-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0367.14001">0367.14001</a>, <q>Ch. III Theorem 12.8</q></cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFGrothendieckDieudonné1963" class="citation journal cs1"><a href="Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexandre</a>; <a href="Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1963). <a rel="nofollow" class="external text" href="http://www.numdam.org/item/PMIHES_1963__17__5_0">"Éléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Seconde partie"</a>. <a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a>. 17: 67. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02684890">10.1007/bf02684890</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0163911">0163911</a>.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/0D4H">"Stacks Project — Lemma 37.30.4"</a>. Stacks.math.columbia.edu. Retrieved 2025-04-27.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/0D4I">"Stacks Project — Lemma 37.30.5"</a>. Stacks.math.columbia.edu. Retrieved 2025-04-27.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFRavi_D.2011" class="citation web cs1">Ravi D., Vakil (2011-06-08). <a rel="nofollow" class="external text" href="https://math216.wordpress.com/2011/06/08/favorite-semicontinuous-functions/">"Favorite semicontinuous functions?"</a>. math216.wordpress.com. Stanford University: Wordpress. Retrieved 2025-04-27.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> Kechris, A. S. (1995). Classical Descriptive Set Theory. Springer.

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> Moschovakis, Y. N. (1980). Descriptive Set Theory. North-Holland.

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> Friedman, H., & Stanley, L. (1989). A Borel reducibility theory for classes of countable structures. J. Symbolic Logic, 54(3), 894–914.

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> Walters, P. (1982). An Introduction to Ergodic Theory. Springer.

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> Glasner, E. (2003). Ergodic Theory via Joinings. American Mathematical Society.

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> Downarowicz, T. (2011). Entropy in Dynamical Systems. Cambridge University Press.

</li>

</ol></div></div>

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

<ul><li><cite id="CITEREFBenesovaKruzik2017" class="citation journal cs1">Benesova, B.; Kruzik, M. (2017). "Weak Lower Semicontinuity of Integral Functionals and Applications". SIAM Review. 59 (4): 703–766. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1601.00390">1601.00390</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F16M1060947">10.1137/16M1060947</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119668631">119668631</a>.</cite></li>

<li><cite id="CITEREFBourbaki1998" class="citation book cs1">Bourbaki, Nicolas (1998). Elements of Mathematics: General Topology, 1–4. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-00636-7</bdi>.</cite></li>

<li><cite id="CITEREFBourbaki1998" class="citation book cs1">Bourbaki, Nicolas (1998). Elements of Mathematics: General Topology, 5–10. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-64563-2</bdi>.</cite></li>

<li><cite id="CITEREFEkelandTémam1999" class="citation cs2">Ekeland; Témam (1999), Convex analysis and variational problems</cite></li>

<li><cite id="CITEREFEngelking1989" class="citation book cs1"><a href="Ryszard_Engelking" title="Ryszard Engelking">Engelking, Ryszard</a> (1989). General Topology. Heldermann Verlag, Berlin. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-88538-006-4</bdi>.</cite></li>

<li><cite id="CITEREFGelbaumOlmsted,_John_M.H.2003" class="citation book cs1">Gelbaum, Bernard R.; Olmsted, John M.H. (2003). Counterexamples in analysis. Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-42875-3</bdi>.</cite></li>

<li><cite id="CITEREFGiusti2003" class="citation cs2">Giusti (2003), Direct methods in the calculus of variations</cite></li>

<li><cite id="CITEREFHyersIsac,_GeorgeRassias,_Themistocles_M.1997" class="citation book cs1">Hyers, Donald H.; Isac, George; Rassias, Themistocles M. (1997). Topics in nonlinear analysis & applications. World Scientific. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-02-2534-2</bdi>.</cite></li>

<li><cite id="CITEREFRockafellar1970" class="citation cs2">Rockafellar (1970), Convex analysis</cite></li>

<li><cite id="CITEREFRockafellarWets1997" class="citation cs2">Rockafellar; Wets (1997), Variational analysis</cite></li>

<li><cite id="CITEREFPhelps1966" class="citation cs2">Phelps (1966), Lectures on the Choquet theorem</cite></li>

<li><cite id="CITEREFStromberg1981" class="citation book cs1">Stromberg, Karl (1981). Introduction to Classical Real Analysis. Wadsworth. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-98012-2</bdi>.</cite></li>

<li><cite id="CITEREFWillard2004" class="citation book cs1">Willard, Stephen (2004) [1970]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-o8xJQ7Ag2cC">General Topology</a>. <a href="Mineola%2C_N.Y." class="mw-redirect" title="Mineola, N.Y.">Mineola, N.Y.</a>: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-43479-7</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/115240">115240</a>.</cite></li>

<li><cite id="CITEREFZălinescu2002" class="citation book cs1">Zălinescu, Constantin (30 July 2002). <a rel="nofollow" class="external text" href="https://archive.org/details/convexanalysisge00zali_934">Convex Analysis in General Vector Spaces</a>. River Edge, N.J. London: <a href="World_Scientific_Publishing" class="mw-redirect" title="World Scientific Publishing">World Scientific Publishing</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-4488-15-0</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1921556">1921556</a>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/285163112">285163112</a> – via <a href="Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Convex_analysis_and_variational_analysis174" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Convex_analysis_and_variational_analysis174" style="font-size:114%;margin:0 4em"><a href="Convex_analysis" title="Convex analysis">Convex analysis</a> and <a href="Variational_analysis" title="Variational analysis">variational analysis</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_combination" title="Convex combination">Convex combination</a></li>

<li><a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Convex_set" title="Convex set">Convex set</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_convexity_topics" title="List of convexity topics">Topics (list)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Convex_geometry" title="Convex geometry">Convex geometry</a></li>

<li><a href="Convex_metric_space" title="Convex metric space">Convex metric space</a></li>

<li><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></li>

<li><a href="Duality_(optimization)" title="Duality (optimization)">Duality</a></li>

<li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector space</a></li>

<li><a href="Simplex" title="Simplex">Simplex</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_conjugate" title="Convex conjugate">Convex conjugate</a></li>

<li><a href="Concave_function" title="Concave function">Concave</a></li>

<li>(<a href="Closed_convex_function" title="Closed convex function">Closed</a></li>

<li><a href="Logarithmically_convex_function" title="Logarithmically convex function">Logarithmically</a></li>

<li><a href="Proper_convex_function" title="Proper convex function">Proper</a></li>

<li><a href="Pseudoconvex_function" title="Pseudoconvex function">Pseudo-</a></li>

<li><a href="Quasiconvex_function" title="Quasiconvex function">Quasi-</a>) <a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Invex_function" title="Invex function">Invex function</a></li>

<li><a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li>

<li><a href="Subderivative" title="Subderivative">Subderivative</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results (list)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carath%C3%A9odory's_theorem_(convex_hull)" title="Carathéodory's theorem (convex hull)">Carathéodory's theorem</a></li>

<li><a href="Ekeland's_variational_principle" title="Ekeland's variational principle">Ekeland's variational principle</a></li>

<li><a href="Fenchel%E2%80%93Moreau_theorem" title="Fenchel–Moreau theorem">Fenchel–Moreau theorem</a></li>

<li><a href="Fenchel-Young_inequality" class="mw-redirect" title="Fenchel-Young inequality">Fenchel-Young inequality</a></li>

<li><a href="Jensen's_inequality" title="Jensen's inequality">Jensen's inequality</a></li>

<li><a href="Hermite%E2%80%93Hadamard_inequality" title="Hermite–Hadamard inequality">Hermite–Hadamard inequality</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman theorem</a></li>

<li><a href="Mazur's_lemma" title="Mazur's lemma">Mazur's lemma</a></li>

<li><a href="Shapley%E2%80%93Folkman_lemma" title="Shapley–Folkman lemma">Shapley–Folkman lemma</a></li>

<li><a href="Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson–Ursescu</a></li>

<li><a href="Ursescu_theorem#Simons'_theorem" title="Ursescu theorem">Simons</a></li>

<li><a href="Ursescu_theorem" title="Ursescu theorem">Ursescu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_hull" title="Convex hull">Convex hull</a></li>

<li>(<a href="Orthogonally_convex_set" class="mw-redirect" title="Orthogonally convex set">Orthogonally</a>, <a href="Pseudoconvexity" title="Pseudoconvexity">Pseudo-</a>) <a href="Convex_set" title="Convex set">Convex set</a></li>

<li><a href="Effective_domain" title="Effective domain">Effective domain</a></li>

<li><a href="Epigraph_(mathematics)" title="Epigraph (mathematics)">Epigraph</a></li>

<li><a href="Hypograph_(mathematics)" title="Hypograph (mathematics)">Hypograph</a></li>

<li><a href="John_ellipsoid" title="John ellipsoid">John ellipsoid</a></li>

<li><a href="Radial_set" title="Radial set">Radial set</a>/<a href="Algebraic_interior" title="Algebraic interior">Algebraic interior</a></li>

<li><a href="Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> (<a href="Convex_series#Types_of_subsets" title="Convex series">(cs, lcs)-closed</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(cs, bcs)-complete</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(lower) ideally convex</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(Hx)</a>, and <a href="Convex_series#Types_of_subsets" title="Convex series">(Hwx)</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Duality_(optimization)" title="Duality (optimization)">Duality</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dual_system" title="Dual system">Dual system</a></li>

<li><a href="Duality_gap" title="Duality gap">Duality gap</a></li>

<li><a href="Strong_duality" title="Strong duality">Strong duality</a></li>

<li><a href="Weak_duality" title="Weak duality">Weak duality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications and related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convexity_in_economics" title="Convexity in economics">Convexity in economics</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-04" href="https://en.wikipedia.org/wiki/?title=Semi-continuity&oldid=1304175428">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>