Set-valued function: Difference between revisions

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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''', (oralso called a '''correspondence''' or '''set-valued [[Relation (mathematics)|relation]]''', is a mathematical [[Function (mathematics)|function]] that maps elements from one set, the [[___domain of a function|___domain of the function]], to subsets of another set.<ref name=":02">{{Cite book |last1=Aliprantis |first1=Charalambos D. |url=https://books.google.com/books?id=Ma31CAAAQBAJ |title=Infinite Dimensional Analysis: A Hitchhiker's Guide |last2=Border |first2=Kim C. |date=2013-03-14 |publisher=Springer Science & Business Media |isbn=978-3-662-03961-8 |pages=523 |language=en}}</ref><ref name=":0">{{Cite book |last1=Wriggers |first1=Peter |url=https://books.google.com/books?id=R4lqCQAAQBAJ |title=New Developments in Contact Problems |last2=Panatiotopoulos |first2=Panagiotis |date=2014-05-04 |publisher=Springer |isbn=978-3-7091-2496-3 |pages=29 |language=en}}</ref> Set-valued functions are used in a variety of mathematical fields, including [[Mathematical optimization|optimization]], [[control theory]] and [[game theory]].

Set-valued functions are also known as [[multivalued functionsfunction]]s in some references,<ref>{{Cite book |last=Repovš |first=Dušan |url=https://www.worldcat.org/oclc/39739641 |title=Continuous selections of multivalued mappings |date=1998 |publisher=Kluwer Academic |others=Pavel Vladimirovič. Semenov |isbn=0-7923-5277-7 |___location=Dordrecht |oclc=39739641}}</ref> but hereinthis article and inthe many others references inarticle [[mathematical analysis]], a [[multivaluedMultivalued function]] is a set-valued function {{mvar|f}} that has a further [[continuous function|continuity]] property, namely thatfollow the choiceauthors ofwho an element in the set <math>f(x)</math> definesmake a corresponding element in each set <math>f(y)</math> for {{mvar|y}} close to {{mvar|x}}, and thus defines [[locally]] an ordinary functiondistinction.

[[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.

== ExamplesExample ==

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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* [[Selection theorem]]

* [[Ursescu theorem]]

* [[Binary relation]]

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