# Characteristic function (convex analysis)

In the field of mathematics known as convex analysis, the **characteristic function** of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function, and one can freely convert between the two, but the characteristic function as defined below is better-suited to the methods of convex analysis.

## Definition

Let be a set, and let be a subset of . The **characteristic function** of is the function

taking values in the extended real number line defined by

## Relationship with the indicator function

Let denote the usual indicator function:

If one adopts the conventions that

- for any , and , except ;
- ; and
- ;

then the indicator and characteristic functions are related by the equations

and

## Bibliography

- Rockafellar, R. T. (1997) [1970].
*Convex Analysis*. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.

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