Characteristic function (convex analysis)
 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 nonmembership) 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 bettersuited to the methods of convex analysis.
Definition
Let X be a set, and let A be a subset of X. The characteristic function of A 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
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 9780691015866.
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