SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Set-valued function</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Set-valued function</span></span>

</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 multi-valued functions of mathematical analysis, see <a href="Multivalued_function" title="Multivalued function">Multivalued function</a>. For functions whose arguments are sets, see <a href="Set_function" title="Set function">Set function</a>.</div>

<p>A <b>set-valued function</b>, also called a <b>correspondence</b> or <b>set-valued <a href="Relation_(mathematics)" title="Relation (mathematics)">relation</a></b>, is a mathematical <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that maps elements from one set, the <a href="Domain_of_a_function" title="Domain of a function">domain of the function</a>, to subsets of another set.<sup id="cite_ref-:02_1-0" class="reference"><a href="#cite_note-:02-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Set-valued functions are used in a variety of mathematical fields, including <a href="Mathematical_optimization" title="Mathematical optimization">optimization</a>, <a href="Control_theory" title="Control theory">control theory</a> and <a href="Game_theory" title="Game theory">game theory</a>.

</p><p>Set-valued functions are also known as <a href="Multivalued_function" title="Multivalued function">multivalued functions</a> in some references,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> but this article and the article <a href="Multivalued_function" title="Multivalued function">Multivalued function</a> follow the authors who make a distinction.

</p>

<div class="mw-heading mw-heading2"><h2 id="Distinction_from_multivalued_functions">Distinction from multivalued functions</h2></div>

<p>Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued functions (which they called <i>set-valued relations</i>) by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a <a href="Function_(mathematics)" title="Function (mathematics)">function</a>.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Geometrically, this means that the graph of a multivalued function is necessarily a line of zero area that doesn't loop, while the graph of a set-valued relation may contain solid filled areas or loops.<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Alternatively, a <a href="Multivalued_function" title="Multivalued function">multivalued function</a> is a set-valued function <span class="texhtml mvar" style="font-style:italic;">f</span> that has a further <a href="Continuous_function" title="Continuous function">continuity</a> property, namely that the choice of an element in the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> defines a corresponding element in each set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./aaa9215c6afa4892692ba05ae4c44f23600ea79d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.243ex; height:2.843ex;" alt="{\displaystyle f(y)}" loading="lazy"></span> for <span class="texhtml mvar" style="font-style:italic;">y</span> close to <span class="texhtml mvar" style="font-style:italic;">x</span>, and thus defines <a href="Locally" class="mw-redirect" title="Locally">locally</a> an ordinary function.

</p>

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

<p>The <a href="Argmax" class="mw-redirect" title="Argmax">argmax</a> of a function is in general, multivalued. For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {argmax} _{x\in \mathbb {R} }\cos(x)=\{2\pi k\mid k\in \mathbb {Z} \}}">

<msub>

<mi>argmax</mi>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {argmax} _{x\in \mathbb {R} }\cos(x)=\{2\pi k\mid k\in \mathbb {Z} \}}</annotation>

</semantics>

</math></span><img src="./9e55617cb0de51b4a2867a898cbd1f3d8ed99248.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.323ex; height:2.843ex;" alt="{\displaystyle \operatorname {argmax} _{x\in \mathbb {R} }\cos(x)=\{2\pi k\mid k\in \mathbb {Z} \}}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Set-valued_analysis">Set-valued analysis</h2></div>

<p><b>Set-valued analysis</b> is the study of sets in the spirit of <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> and <a href="General_topology" title="General topology">general topology</a>.

</p><p>Instead of considering collections of only points, set-valued analysis considers collections of sets. If a collection of sets is endowed with a topology, or inherits an appropriate topology from an underlying topological space, then the convergence of sets can be studied.

</p><p>Much of set-valued analysis arose through the study of <a href="Mathematical_economics" title="Mathematical economics">mathematical economics</a> and <a href="Optimal_control" title="Optimal control">optimal control</a>, partly as a generalization of <a href="Convex_analysis" title="Convex analysis">convex analysis</a>; the term "<a href="Variational_analysis" title="Variational analysis">variational analysis</a>" is used by authors such as <a href="R._Tyrrell_Rockafellar" title="R. Tyrrell Rockafellar">R. Tyrrell Rockafellar</a> and <a href="Roger_J-B_Wets" title="Roger J-B Wets">Roger J-B Wets</a>, <a href="Jonathan_Borwein" title="Jonathan Borwein">Jonathan Borwein</a> and <a href="Adrian_Lewis" title="Adrian Lewis">Adrian Lewis</a>, and <a href="Boris_Mordukhovich" title="Boris Mordukhovich">Boris Mordukhovich</a>. In optimization theory, the convergence of approximating <a href="Subdifferential" class="mw-redirect" title="Subdifferential">subdifferentials</a> to a subdifferential is important in understanding necessary or sufficient conditions for any minimizing point.

</p><p>There exist set-valued extensions of the following concepts from point-valued analysis: <a href="Continuous_(mathematics)" class="mw-redirect" title="Continuous (mathematics)">continuity</a>, <a href="Differentiation_(mathematics)" class="mw-redirect" title="Differentiation (mathematics)">differentiation</a>, <a href="Integral" title="Integral">integration</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a>, <a href="Contraction_mapping" title="Contraction mapping">contraction mappings</a>, <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, <a href="Fixed-point_theorem" title="Fixed-point theorem">fixed-point theorems</a>,<sup id="cite_ref-kakutani_5-0" class="reference"><a href="#cite_note-kakutani-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimization</a>, and <a href="Topological_degree_theory" title="Topological degree theory">topological degree theory</a>. In particular, <a href="Equation" title="Equation">equations</a> are generalized to <a href="Inclusion_(set_theory)" class="mw-redirect" title="Inclusion (set theory)">inclusions</a>, while differential equations are generalized to <a href="Differential_inclusion" title="Differential inclusion">differential inclusions</a>.

</p><p>One can distinguish multiple concepts generalizing <a href="Continuity_(mathematics)" class="mw-redirect" title="Continuity (mathematics)">continuity</a>, such as the <a href="Closed_graph" class="mw-redirect" title="Closed graph">closed graph</a> property and <a href="Hemicontinuity" title="Hemicontinuity">upper and lower hemicontinuity</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup>. There are also various generalizations of <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> to multifunctions.

</p>

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

<p>Set-valued functions arise in <a href="Optimal_control" title="Optimal control">optimal control theory</a>, especially <a href="Differential_inclusion" title="Differential inclusion">differential inclusions</a> and related subjects as <a href="Game_theory" title="Game theory">game theory</a>, where the <a href="Kakutani_fixed-point_theorem" title="Kakutani fixed-point theorem">Kakutani fixed-point theorem</a> for set-valued functions has been applied to prove existence of <a href="Nash_equilibrium" title="Nash equilibrium">Nash equilibria</a>. This among many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity.

</p><p>Nevertheless, lower semi-continuous multifunctions usually possess continuous selections as stated in the <a href="Michael_selection_theorem" title="Michael selection theorem">Michael selection theorem</a>, which provides another characterisation of <a href="Paracompact" class="mw-redirect" title="Paracompact">paracompact</a> spaces.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Other selection theorems, like Bressan-Colombo directional continuous selection, <a href="Kuratowski_and_Ryll-Nardzewski_measurable_selection_theorem" title="Kuratowski and Ryll-Nardzewski measurable selection theorem">Kuratowski and Ryll-Nardzewski measurable selection theorem</a>, Aumann measurable selection, and Fryszkowski selection for decomposable maps are important in <a href="Optimal_control" title="Optimal control">optimal control</a> and the theory of <a href="Differential_inclusion" title="Differential inclusion">differential inclusions</a>.

</p>

<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 reflist-lower-alpha">

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Some authors use the term ‘semicontinuous’ instead of ‘hemicontinuous’.</span>

</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="CITEREFAliprantisBorder2013" class="citation book cs1">Aliprantis, Charalambos D.; Border, Kim C. (2013-03-14). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ma31CAAAQBAJ"><i>Infinite Dimensional Analysis: A Hitchhiker's Guide</i></a>. Springer Science & Business Media. p. 523. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-662-03961-8</bdi>.</cite></span>

</li>

<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFWriggersPanatiotopoulos2014" class="citation book cs1">Wriggers, Peter; Panatiotopoulos, Panagiotis (2014-05-04). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R4lqCQAAQBAJ"><i>New Developments in Contact Problems</i></a>. Springer. p. 29. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7091-2496-3</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRepovš1998" class="citation book cs1">Repovš, Dušan (1998). <i>Continuous selections of multivalued mappings</i>. Pavel Vladimirovič. Semenov. Dordrecht: Kluwer Academic. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-5277-7</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/39739641">39739641</a>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFAumann1965" class="citation journal cs1"><a href="Robert_Aumann" title="Robert Aumann">Aumann, Robert J.</a> (1965). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-247X%2865%2990049-1">"Integrals of Set-Valued Functions"</a>. <i><a href="Journal_of_Mathematical_Analysis_and_Applications" title="Journal of Mathematical Analysis and Applications">Journal of Mathematical Analysis and Applications</a></i>. <b>12</b> (1): <span class="nowrap">1–</span>12. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-247X%2865%2990049-1">10.1016/0022-247X(65)90049-1</a></span>.</cite></span>

</li>

<li id="cite_note-kakutani-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-kakutani_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKakutani1941" class="citation journal cs1"><a href="Shizuo_Kakutani" title="Shizuo Kakutani">Kakutani, Shizuo</a> (1941). "A generalization of Brouwer's fixed point theorem". <i><a href="Duke_Mathematical_Journal" title="Duke Mathematical Journal">Duke Mathematical Journal</a></i>. <b>8</b> (3): <span class="nowrap">457–</span>459. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-41-00838-4">10.1215/S0012-7094-41-00838-4</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFErnest_Michael1956" class="citation journal cs1"><a href="Ernest_Michael" title="Ernest Michael">Ernest Michael</a> (Mar 1956). <a rel="nofollow" class="external text" href="http://www.renyi.hu/~descript/papers/Michael_1.pdf">"Continuous Selections. I"</a> <span class="cs1-format">(PDF)</span>. <i>Annals of Mathematics</i>. Second Series. <b>63</b> (2): <span class="nowrap">361–</span>382. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969615">10.2307/1969615</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F119700">10338.dmlcz/119700</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969615">1969615</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFDušan_RepovšP.V._Semenov2008" class="citation journal cs1"><a href="Du%C5%A1an_Repov%C5%A1" title="Dušan Repovš">Dušan Repovš</a>; P.V. Semenov (2008). "Ernest Michael and theory of continuous selections". <i>Topology Appl</i>. <b>155</b> (8): <span class="nowrap">755–</span>763. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0803.4473">0803.4473</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.topol.2006.06.011">10.1016/j.topol.2006.06.011</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:14509315">14509315</a>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>K. Deimling, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=D9pgTAujcKcC">Multivalued Differential Equations</a></i>, Walter de Gruyter, 1992</li>

<li>C. D. Aliprantis and K. C. Border, <i>Infinite dimensional analysis. Hitchhiker's guide</i>, Springer-Verlag Berlin Heidelberg, 2006</li>

<li>J. Andres and L. Górniewicz, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=PanqCAAAQBAJ&q=multivalued">Topological Fixed Point Principles for Boundary Value Problems</a></i>, Kluwer Academic Publishers, 2003</li>

<li>J.-P. Aubin and A. Cellina, <i>Differential Inclusions, Set-Valued Maps And Viability Theory</i>, Grundl. der Math. Wiss. 264, Springer - Verlag, Berlin, 1984</li>

<li>J.-P. Aubin and <a href="H%C3%A9l%C3%A8ne_Frankowska" title="Hélène Frankowska">H. Frankowska</a>, <i>Set-Valued Analysis</i>, Birkhäuser, Basel, 1990</li>

<li><a href="Du%C5%A1an_Repov%C5%A1" title="Dušan Repovš">D. Repovš</a> and P.V. Semenov, <a rel="nofollow" class="external text" href="https://www.springer.com/gp/book/9780792352778?cm_mmc=sgw-_-ps-_-book-_-0-7923-5277-7"><i>Continuous Selections of Multivalued Mappings</i></a>, Kluwer Academic Publishers, Dordrecht 1998</li>

<li>E. U. Tarafdar and M. S. R. Chowdhury, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Cir88lF64xIC"><i>Topological methods for set-valued nonlinear analysis</i></a>, World Scientific, Singapore, 2008</li>

<li><cite id="CITEREFMitroiNikodemWąsowicz2013" class="citation journal cs1">Mitroi, F.-C.; Nikodem, K.; Wąsowicz, S. (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fdema-2013-0483">"Hermite-Hadamard inequalities for convex set-valued functions"</a>. <i>Demonstratio Mathematica</i>. <b>46</b> (4): <span class="nowrap">655–</span>662. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fdema-2013-0483">10.1515/dema-2013-0483</a></span>.</cite></li></ul>

<ul><li><a href="Selection_theorem" title="Selection theorem">Selection theorem</a></li>

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

<li><a href="Binary_relation" title="Binary relation">Binary relation</a></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="Function330" 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="Function330" style="font-size:114%;margin:0 4em"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_the_function_concept" title="History of the function concept">History</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul>

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

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

<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

<li><a href="Linear_map" title="Linear map">Linear</a></li>

<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>

<li><a href="Rational_function" title="Rational function">Rational</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

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

<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>

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

<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>

<li><a href="Function_space" title="Function space">Space</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

<li><a href="Morphism" title="Morphism">Morphism</a></li>

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

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</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 2025-07-18" href="https://en.wikipedia.org/wiki/?title=Set-valued_function&oldid=1301208119">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>