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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 (mathematics)|function]]''',<ref name=":0">{{Cite book |last=Aliprantis |first=Charalambos D. |url=https://www.google.com.br/books/edition/Infinite_Dimensional_Analysis/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> also called a '''correspondence'''<ref name=":0" /> or '''set-valued [[Relation (mathematics)|relation]]''',<ref>{{Cite book |last=Wriggers |first=Peter |url=https://www.google.com.br/books/edition/New_Developments_in_Contact_Problems/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> is a mathematical function that maps elements from one set, the [[___domain of a function|___domain of the function]], to subsets of another set.<ref name=":0" /> 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 function|multivalued functions]] 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
▲[[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 relations (also called [[Set-valued function|set-valued functions]]) by the fact that multivalued functions only take multiple values at denumerably many points, and otherwise behave like a [[Function (mathematics)|function]].<ref name=":0">{{Cite book |last=Wriggers |first=Peter |url=https://www.google.com.br/books/edition/New_Developments_in_Contact_Problems/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> 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 relations may contain solid filled areas or loops.<ref name=":0" />
Alternatively, [[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.
== Examples ==
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