[[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''', also 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 |lastlast1=Aliprantis |firstfirst1=Charalambos D. |url=https://wwwbooks.google.com.br/books/edition/Infinite_Dimensional_Analysis/?id=Ma31CAAAQBAJ |title=Infinite Dimensional Analysis: A Hitchhiker’sHitchhiker'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 |lastlast1=Wriggers |firstfirst1=Peter |url=https://wwwbooks.google.com.br/books/edition/New_Developments_in_Contact_Problems/?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 function]]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 this article and the article [[Multivalued function]] follow the authors who make a distinction.
