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.
Let [math]\displaystyle{ X }[/math] be a set, and let [math]\displaystyle{ A }[/math] be a subset of [math]\displaystyle{ X }[/math]. The characteristic function of [math]\displaystyle{ A }[/math] is the function
taking values in the extended real number line defined by
Let [math]\displaystyle{ \mathbf{1}_{A} : X \to \mathbb{R} }[/math] 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 [math]\displaystyle{ \chi_{A} (x) }[/math] for a set [math]\displaystyle{ A }[/math] is the tangent cone of that set in [math]\displaystyle{ x }[/math].
![]() | Original source: https://en.wikipedia.org/wiki/Characteristic function (convex analysis).
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