Support (mathematics) - Compact Support

Functions with compact support in X are those with support that is a compact subset of X. For example, if X is the real line, they are functions of bounded support and therefore vanish at infinity (and negative infinity).

Real-valued compactly supported smooth functions on a Euclidean space are called bump functions. Mollifiers are an important special case of bump functions as they can be used in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.

In good cases, functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. As an intuition for more complex examples, and in the language of limits, for any ε > 0, any function f on the real line R that vanishes at infinity can be approximated by choosing an appropriate compact subset C of R such that

for all x ∈ X, where is the indicator function of C. Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact.