Multivalued function

It has been suggested that this article be merged into Relation (mathematics). (Discuss) Proposed since March 2024.

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (January 2020) (Learn how and when to remove this message)

In mathematics, a multivalued function (also known as a multiple-valued function) is a function that has two or more values in its range for at least one point in its domain.^[1] It is a set-valued function with additional properties depending on context. The terms multifunction and many-valued function are sometimes also used.

A multivalued function of sets f : X → Y is a subset

${\displaystyle \Gamma _{f}\ \subseteq \ X\times Y.}$

Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γ_f. If f is an ordinary function, it is a multivalued function by taking its graph

${\displaystyle \Gamma _{f}\ =\ \{(x,f(x))\ :\ x\in X\}.}$

They are called single-valued functions to distinguish them.