Compact support

A function has compact support when its support is a compact set. On the real line, this means the closure of its nonzero region is closed and bounded. The function $f(x)=\max (0,1-|x|)$ has compact support $[-1,1]$.

Decay is a different condition: $g(x)=1/(1+x^2)$ tends to zero as $|x|\to \infty$, but is nonzero everywhere, so its support is all of $\mathbb{R}$ and is not compact. Smooth functions with compact support are used as test functions in distribution theory.