> [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]] and $f \in C(X)$ be a real/complex-valued [[Space of Continuous Functions|continuous function]]. $f$ is **compactly supported** if its [[Support of Function|support]] is [[Compactness|compact]]. > [!definition] > > Let $X$ be a topological space, then > $ > C_c(X) = \bracs{f \in C(X): \supp{f} \text{ compact}} > $ > is the space of all **compactly supported** functions. > [!theorem] > > Every compactly supported function [[Vanishes at Infinity|vanishes at infinity]]. > [!theorem] > > If $f \in C_c(\real^n)$, then $f$ is [[Uniform Continuity|uniformly continuous]]. > > *Proof*. Let $\varepsilon > 0$. For any $x \in \supp{f}$, there exists $\delta_x > 0$ such that $\abs{f(x - y) - f(x)} < \frac{1}{2}\varepsilon$ if $\abs{y} < \delta_x$ (in other words $\abs{f(y) - f(z)} < \varepsilon$ for all $y, z \in B(x, \delta_x)$). This forms an [[Open Cover|open cover]] of $\supp{f}$. Since $\supp{f}$ is compact, there exists a finite subcover. The desired $\delta$ is the minimum of all relevant $\delta_x$s.