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{{About|set-valued functions as considered in variational analysis|multi-valued functions as they are considered in complex analysis|multivalued function}}
A '''set-valued function''' (or '''correspondence''') is a mathematical function that maps elements from one set, known as the ___domain, to sets of elements in another set. Set-valued functions are used in a variety of mathematical fields, including [[Mathematical optimization|optimization]], [[control theory]] and [[game theory]].
== Examples ==▼
The [[argmax]] of a function is in general, multivalued. For example, <math>\operatorname{argmax}_{x \in \mathbb{R}} \cos(x) = \{2 \pi k\mid k \in \mathbb{Z}\}</math>.▼
== Set-valued analysis ==
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There exist set-valued extensions of the following concepts from point-valued analysis: [[Continuous (mathematics)|continuity]], [[Differentiation (mathematics)|differentiation]], [[Integral|integration]],<ref>{{cite journal |last=Aumann |first=Robert J. |author-link=Robert Aumann |year=1965 |title=Integrals of Set-Valued Functions |journal=[[Journal of Mathematical Analysis and Applications]] |volume=12 |issue=1 |pages=1–12 |doi=10.1016/0022-247X(65)90049-1 |doi-access=free}}</ref> [[implicit function theorem]], [[Contraction mapping|contraction mappings]], [[measure theory]], [[Fixed-point theorem|fixed-point theorems]],<ref name="kakutani">{{cite journal |last=Kakutani |first=Shizuo |author-link=Shizuo Kakutani |year=1941 |title=A generalization of Brouwer's fixed point theorem |journal=[[Duke Mathematical Journal]] |volume=8 |issue=3 |pages=457–459 |doi=10.1215/S0012-7094-41-00838-4}}</ref> [[Optimization (mathematics)|optimization]], and [[topological degree theory]]. In particular, [[Equation|equations]] are generalized to [[Inclusion (set theory)|inclusions]], while differential equations are generalized to [[Differential inclusion|differential inclusions]].
▲== Examples ==
▲The [[argmax]] of a function is in general, multivalued. For example, <math>\operatorname{argmax}_{x \in \mathbb{R}} \cos(x) = \{2 \pi k\mid k \in \mathbb{Z}\}</math>.
One can distinguish multiple concepts generalizing [[Continuity (mathematics)|continuity]], such as the [[closed graph]] property and [[Hemicontinuity|upper and lower hemicontinuity]]{{efn|Some authors use the term ‘semicontinuous’ instead of ‘hemicontinuous’.}}. There are also various generalizations of [[Measure (mathematics)|measure]] to multifunctions.
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