Multivalued function

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. 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.