Continuous Functions with Compact Support: the Zero Function, Closure under Operations, Vanishing off the Support, Existence of the Maximum Modulus, and Uniform Continuity

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $(X,d)$ be a metric space.

(i) Zero. The constant function $0$ on $X$ is continuous and has compact support: its support is empty.

Let now $f,g:X\to\mathbb{R}$ be continuous functions with compact support, and let $c\in\mathbb{R}$.

(ii) Operations. The functions $f+g$, $cf$, $fg$, $|f|$, $\max(f,g)$ and $\min(f,g)$ are continuous and have compact support, with

$$\operatorname{supp}(f+g),\ \operatorname{supp}\max(f,g),\ \operatorname{supp}\min(f,g)\subseteq\operatorname{supp}f\cup\operatorname{supp}g,\qquad\operatorname{supp}(cf)\subseteq\operatorname{supp}f,$$ $$\operatorname{supp}(fg)\subseteq\operatorname{supp}f\cap\operatorname{supp}g,\qquad\operatorname{supp}|f|=\operatorname{supp}f.$$

(iii) Vanishing off the support. $f(x)=0$ for every $x\in X\setminus\operatorname{supp}f$.

(iv) Maximum. The subset $\{0\}\cup\{|f(x)|:x\in X\}$ of $\mathbb{R}$ has a greatest element.

(v) Uniform continuity. $f$ is uniformly continuous.