Set-valued function: Difference between revisions

Revision as of 23:37, 10 February 2024

Revision as of 18:09, 25 October 2024

Line 2:

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

A '''set-valued [[Function (mathematics)|function]]''',<ref (orname=":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 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.

[[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''.]]