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 Subgradient The subgradient of for a set is the tangent cone of that set in . Bibliography Convex analysis