Set-valued function: Difference between revisions

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{{About||multi-valued functions of mathematical analysis|Multivalued function|functions whose arguments are sets|Set function}}

A '''set-valued function''' (or '''correspondence''') is a mathematical function that maps elements from one set, the [[___domain of a function|___domain of the function]], to subsets of {{Functions}}another set. Set-valued functions are used in a variety of mathematical fields, including [[Mathematical optimization|optimization]], [[control theory]] and [[game theory]].

InSet-valued functions are also known as 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 herein and in many others references in [[mathematical analysis]], 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.

Set-valued functions are used in a variety of mathematical fields, including [[Mathematical optimization|optimization]], [[control theory]] and [[game theory]].

[[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''.]]

In [[mathematical analysis]], 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.

== Examples ==