Compactly supported function

A function $f:U\to\mathbb R^m$ is compactly supported in $U$ when its support

\operatorname{supp}_U f=\overline{\{x\in U:f(x)\ne0\}}^{\,U}

is a compact subset of $U$ . For open $U\subseteq\mathbb R^n$ , the notation $C_c^k(U;\mathbb R^m)$ denotes compactly supported class $C^k$ maps, and $C_c^\infty$ denotes the smooth ones.

The domain matters

If $U=(0,1)$ , the constant function $1$ has bounded support but does not have compact support in $U$ . Compact support in an open Euclidean domain leaves a positive distance from its boundary. A smooth function with such support extends smoothly by zero to $\mathbb R^n$ , since it already vanishes in a neighborhood of each boundary point.

Relation to bump functions

A bump function is a smooth compactly supported scalar function. Compact support itself is a condition on support and does not imply continuity or differentiability.