Multivalued function

In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, 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; though some authors do not distinguish between set-valued functions and multifunctions.

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

Γ

f

⊆

X

×

Y

.

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

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

Γ

f

=

{

(

x

,

f

(

x

)

)

:

x

∈

X

}

.

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

They are called single-valued functions to distinguish them.

Motivation

The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function

f

(

z

)

{\displaystyle f(z)}

in some neighbourhood of a point

z

=

a

{\displaystyle z=a}

. This is the case for functions defined by the implicit function theorem or by a Taylor series around

z

=

a

{\displaystyle z=a}

. In such a situation, one may extend the domain of the single-valued function

f

(

z

)

{\displaystyle f(z)}

along curves in the complex plane starting at

a

{\displaystyle a}

.