@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-K0PQKG10-G
  skos:prefLabel "calcul différentiel"@fr, "differential calculus"@en ;
  a skos:Concept ;
  skos:narrower psr:-S3DNT3VV-M .

psr:-SKTRS1V0-R
  skos:prefLabel "real analysis"@en, "analyse réelle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-S3DNT3VV-M .

psr:-S3DNT3VV-M
  dc:created "2023-08-02"^^xsd:date ;
  skos:broader psr:-K0PQKG10-G, psr:-SKTRS1V0-R ;
  skos:definition """In calculus, the <b>extreme value theorem</b> states that if a real-valued function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>f</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> is continuous on the closed interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle [a,b]}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">]</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle [a,b]}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="[a,b]"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>f</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> must attain a maximum and a minimum, each at least once. That is, there exist numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle c}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>c</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle c}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="c"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle d}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>d</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle d}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="d"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle [a,b]}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">]</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle [a,b]}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="[a,b]"></span> such that:
         <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle f(c)\\\\geq f(x)\\\\geq f(d)\\\\quad \\orall x\\\\in [a,b]}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>f</mi>
         <mo stretchy="false">(</mo>
         <mi>c</mi>
         <mo stretchy="false">)</mo>
         <mo>≥<!-- ≥ --></mo>
         <mi>f</mi>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mo>≥<!-- ≥ --></mo>
         <mi>f</mi>
         <mo stretchy="false">(</mo>
         <mi>d</mi>
         <mo stretchy="false">)</mo>
         <mspace width="1em"></mspace>
         <mi mathvariant="normal">∀<!-- ∀ --></mi>
         <mi>x</mi>
         <mo>∈<!-- ∈ --></mo>
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">]</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle f(c)\\\\geq f(x)\\\\geq f(d)\\\\quad \\orall x\\\\in [a,b]}</annotation>
         </semantics>
         </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36b6b81ed80475cc03d569abd175007cba405615" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.353ex; height:2.843ex;" alt="{\\\\displaystyle f(c)\\\\geq f(x)\\\\geq f(d)\\\\quad \\orall x\\\\in [a,b]}"></div>
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Extreme_value_theorem">https://en.wikipedia.org/wiki/Extreme_value_theorem</a>)"""@en, """En mathématiques, et plus précisément en analyse réelle, le théorème des valeurs extrêmes ou théorème des bornes atteintes ou théorème des bornes ou théorème de Weierstrass énonce qu'une fonction continue sur un segment est bornée et atteint ses bornes. Autrement dit, une telle fonction possède un minimum et un maximum sur ce segment. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_valeurs_extr%C3%AAmes">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_valeurs_extr%C3%AAmes</a>)"""@fr ;
  dc:modified "2023-08-02"^^xsd:date ;
  skos:altLabel "théorème des bornes"@fr, "théorème des bornes atteintes"@fr, "théorème de Weierstrass"@fr ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:prefLabel "extreme value theorem"@en, "théorème des valeurs extrêmes"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_valeurs_extr%C3%AAmes>, <https://en.wikipedia.org/wiki/Extreme_value_theorem> .