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[[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''.]]
A '''set-valued function''', also called a '''correspondence''' or '''set-valued [[Relation (mathematics)|relation]]''', is a mathematical [[Function (mathematics)|function]] that maps elements from one set, the [[___domain of a function|___domain of the function]], to subsets of another set.<ref name=":02">{{Cite book |
Set-valued functions are also known as [[multivalued function]]s in some references,<ref>{{Cite book |last=Repovš |first=Dušan
== Distinction from multivalued functions ==
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Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued functions (which they called ''set-valued relations'') by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a [[Function (mathematics)|function]].<ref name=":0" /> Geometrically, this means that the graph of a multivalued function is necessarily a line of zero area that doesn't loop, while the graph of a set-valued relation may contain solid filled areas or loops.<ref name=":0" />
Alternatively, 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.
== Example ==
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