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{{Short description|Function whose values are sets (mathematics)}}
{{About||multi-valued functions of mathematical analysis|Multivalued function|functions whose arguments are sets|Set function}}
[[File:Multivalued_function.svg|right|frame|This diagram represents a multi-valued, but not a proper (single-valued) [[Function (mathematics)|function]], because the element 3 in ''X'' is associated with two elements, ''b'' and ''c'', in ''Y''.]]▼
A '''set-valued function''',
Set-valued functions are also known as [[multivalued
▲[[File:Multivalued_function.svg|right|frame|This diagram represents a multi-valued, but not a proper (single-valued) [[Function (mathematics)|function]], because the element 3 in ''X'' is associated with two elements, ''b'' and ''c'', in ''Y''.]]
== Distinction from multivalued functions ==
[[File:Multivalued_functions_illustration.svg|thumb|right|600px|Illustration distinguishing multivalued functions from set-valued relations according to the criterion in page 29 of ''New Developments in Contact Problems'' by Wriggers and Panatiotopoulos (2014).]]
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 ''set-valued relations'') by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a [[Function (mathematics)|function]].<ref name=":0" /> 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.<ref name=":0" />
Alternatively, a [[multivalued function]] is a set-valued function {{mvar|f}} that has a further [[continuous function|continuity]] property, namely that the choice of an element in the set <math>f(x)</math> defines a corresponding element in each set <math>f(y)</math> for {{mvar|y}} close to {{mvar|x}}, and thus defines [[locally]] an ordinary function.
==
The [[argmax]] of a function is in general, multivalued. For example, <math>\operatorname{argmax}_{x \in \mathbb{R}} \cos(x) = \{2 \pi k\mid k \in \mathbb{Z}\}</math>.
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Set-valued functions arise in [[Optimal control|optimal control theory]], especially [[Differential inclusion|differential inclusions]] and related subjects as [[game theory]], where the [[Kakutani fixed-point theorem]] for set-valued functions has been applied to prove existence of [[Nash equilibrium|Nash equilibria]]. 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.
Nevertheless, lower semi-continuous multifunctions usually possess continuous selections as stated in the [[Michael selection theorem]], which provides another characterisation of [[paracompact]] spaces.<ref>{{cite journal |author=Ernest Michael |author-link=Ernest Michael |date=Mar 1956 |title=Continuous Selections. I |url=http://www.renyi.hu/~descript/papers/Michael_1.pdf |journal=Annals of Mathematics |series=Second Series |volume=63 |pages=361–382 |doi=10.2307/1969615 |jstor=1969615 |number=2 |hdl=10338.dmlcz/119700}}</ref><ref>{{cite journal |author1=Dušan Repovš |author1-link=Dušan Repovš |author2=P.V. Semenov |year=2008 |title=Ernest Michael and theory of continuous selections |journal=Topology Appl. |volume=155 |pages=755–763 |arxiv=0803.4473 |doi=10.1016/j.topol.2006.06.011 |number=8|s2cid=14509315 }}</ref> Other selection theorems, like Bressan-Colombo directional continuous selection, [[Kuratowski and Ryll-Nardzewski measurable selection theorem]], Aumann measurable selection, and Fryszkowski selection for decomposable maps are important in [[optimal control]] and the theory of [[Differential inclusion|differential inclusions]].
== Notes ==
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== Further reading ==
* K. Deimling, ''[https://books.google.com/books?id=D9pgTAujcKcC
* C. D. Aliprantis and K. C. Border, ''Infinite dimensional analysis. Hitchhiker's guide'', Springer-Verlag Berlin Heidelberg, 2006
* J. Andres and L. Górniewicz, ''[https://books.google.com/books?id=PanqCAAAQBAJ
* J.-P. Aubin and A. Cellina, ''Differential Inclusions, Set-Valued Maps And Viability Theory'', Grundl. der Math. Wiss. 264, Springer - Verlag, Berlin, 1984
* J.-P. Aubin and [[Hélène Frankowska|H. Frankowska]], ''Set-Valued Analysis'', Birkhäuser, Basel, 1990
* [[Dušan Repovš|D. Repovš]] and P.V. Semenov, [https://www.springer.com/gp/book/9780792352778?cm_mmc=sgw-_-ps-_-book-_-0-7923-5277-7 ''Continuous Selections of Multivalued Mappings''], Kluwer Academic Publishers, Dordrecht 1998
* E. U. Tarafdar and M. S. R. Chowdhury, [https://books.google.com/books?id=Cir88lF64xIC ''Topological methods for set-valued nonlinear analysis''], World Scientific, Singapore, 2008
* {{cite journal |
== See also ==
* [[Selection theorem]]
* [[Ursescu theorem]]
* [[Binary relation]]
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