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.
is the function taking values in the extended real number line defined by Let
denote the usual indicator function: If one adopts the conventions that then the indicator and characteristic functions are related by the equations and The subgradient of
is the tangent cone of that set in