Support ($\mathrm{supp}f$)

For a scalar-valued function $f$ on a topological space $X$, its support isThe closure matters. On the real line, $f(x)=\max (0,1-|x|)$ is nonzero exactly on $(-1,1)$, but its support is the closed interval $[-1,1]$. A point where $f$ vanishes can still belong to the support if nonzero values occur arbitrarily near it.

For the discrete topology, every subset is closed, so the support is simply the set of inputs with nonzero values. This is the convention used when representing a finitely supported count function. A function has compact support when the displayed closed set is compact.

Suppose functions $f,g$ vanish outside finite sets $S_f,S_g$, and a binary operation $F$ satisfies $F(0,0)=0$. The pointwise operation result $h(x)=F(f(x),g(x))$ then vanishes outside the union $S_f\cup S_g$: elsewhere both inputs are zero. If these sets are their supports, this proves the displayed support inclusion.

Enumerating the union is sufficient for addition, subtraction, pointwise maximum and pointwise minimum of signed count functions. It also suffices to test their pointwise order, since $0\le 0$ elsewhere. Taking only the intersection can miss a nonzero value present on just one side. Scanning only the left support can also miss a negative right-side value when checking $f\le g$.

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.