Set-valued function

A set-valued function (or correspondence) is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including optimization, control theory and game theory.

Set-valued functions are also known as multivalued functions in some references,^[1] but herein and in many others references in mathematical analysis, a multivalued function is a set-valued function f that has a further continuity property, namely that the choice of an element in the set ${\displaystyle f(x)}$ defines a corresponding element in each set ${\displaystyle f(y)}$ for y close to x, and thus defines locally an ordinary function.