Multivalued function

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In mathematics, a multivalued function,^[1] multiple-valued function,^[2] many-valued function,^[3] or multifunction,^[4] is a function that has two or more values in its range for at least one point in its domain.^[5] It is a set-valued function with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions,^[6] but English Wikipedia currently does, having a separate article for each.

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.