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Tito Omburo (talk | contribs) →Motivation: {{Main|Global analytic function}} |
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[[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 |url-access=subscription }}</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
A ''multivalued function'' of sets ''f : X → Y'' is a subset
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== Motivation ==
{{Main|Global analytic function}}
The term multivalued function originated in complex analysis, from [[analytic continuation]]. It often occurs that one knows the value of a complex [[analytic function]] <math>f(z)</math> in some [[neighbourhood (mathematics)|neighbourhood]] of a point <math>z=a</math>. This is the case for functions defined by the [[implicit function theorem]] or by a [[Taylor series]] around <math>z=a</math>. In such a situation, one may extend the ___domain of the single-valued function <math>f(z)</math> along curves in the complex plane starting at <math>a</math>. In doing so, one finds that the value of the extended function at a point <math>z=b</math> depends on the chosen curve from <math>a</math> to <math>b</math>; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.
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