Editing Multivalued function (section)

{{Short description|Generalized mathematical function}}
{{More footnotes needed|date=January 2020}}
{{About|multivalued functions as they are considered in mathematical analysis|set-valued functions as considered in variational analysis|set-valued function}}{{distinguish|Multivariate function}}

[[File:Multivalued_function.svg|thumb|Multivalued function {1,2,3} → {a,b,c,d}.]]
In [[mathematics]], a '''multivalued function''',<ref>{{Cite web |title=Multivalued Function |url=https://archive.lib.msu.edu/crcmath/math/math/m/m450.htm |access-date=2024-10-25 |website=archive.lib.msu.edu}}</ref> '''multiple-valued function''',<ref>{{Cite web |title=Multiple Valued Functions {{!}} Complex Variables with Applications {{!}} Mathematics |url=https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-fall-1999/pages/study-materials/multiple-valued-functions/ |access-date=2024-10-25 |website=MIT OpenCourseWare |language=en}}</ref> '''many-valued function''',<ref>{{Cite journal |last1=Al-Rabadi |first1=Anas |last2=Zwick |first2=Martin |date=2004-01-01 |title=Modified Reconstructability Analysis for Many-Valued Functions and Relations |url=https://pdxscholar.library.pdx.edu/sysc_fac/30/ |journal=Kybernetes |volume=33 |issue=5/6 |pages=906–920 |doi=10.1108/03684920410533967}}</ref> or '''multifunction''',<ref>{{Cite journal |last1=Ledyaev |first1=Yuri |last2=Zhu |first2=Qiji |date=1999-09-01 |title=Implicit Multifunction Theorems |url=https://scholarworks.wmich.edu/math_pubs/22/ |journal=Set-Valued Analysis Volume |volume=7 |issue=3 |pages=209–238|doi=10.1023/A:1008775413250 }}</ref> is a function that has two or more values in its range for at least one point in its domain.<ref>{{cite web |title=Multivalued Function |url=https://mathworld.wolfram.com/MultivaluedFunction.html |website=Wolfram MathWorld |access-date=10 February 2024}}</ref> It is a [[set-valued function]] with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions,<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 English Wikipedia currently does, having a separate article for each. 

A ''multivalued function'' of sets ''f : X → Y'' is a subset 
:<math> \Gamma_f\ \subseteq \ X\times Y.</math>
Write ''f(x)'' for the set of those ''y'' ∈ ''Y'' with (''x,y'') ∈ ''Γ<sub>f</sub>''. If ''f'' is an ordinary function, it is a multivalued function by taking its [[Graph of a function|graph]]
:<math> \Gamma_f\ =\ \{(x,f(x))\ :\ x\in X\}.</math>
They are called '''single-valued functions''' to distinguish them.