Set-valued function

[2] 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.

Much of set-valued analysis arose through the study of mathematical economics and optimal control, partly as a generalization of convex analysis; the term "variational analysis" is used by authors such as R. Tyrrell Rockafellar and Roger J-B Wets, Jonathan Borwein and Adrian Lewis, and Boris Mordukhovich.

There exist set-valued extensions of the following concepts from point-valued analysis: continuity, differentiation, integration,[4] implicit function theorem, contraction mappings, measure theory, fixed-point theorems,[5] optimization, and topological degree theory.

One can distinguish multiple concepts generalizing continuity, such as the closed graph property and upper and lower hemicontinuity[a].

Set-valued functions arise in optimal control theory, especially differential inclusions and related subjects as game theory, where the Kakutani fixed-point theorem for set-valued functions has been applied to prove existence of Nash equilibria.

Nevertheless, lower semi-continuous multifunctions usually possess continuous selections as stated in the Michael selection theorem, which provides another characterisation of paracompact spaces.

This diagram represents a multi-valued, but not a proper (single-valued) function , because the element 3 in X is associated with two elements, b and c , in Y .
Illustration distinguishing multivalued functions from set-valued relations according to the criterion in page 29 of New Developments in Contact Problems by Wriggers and Panatiotopoulos (2014).