Suppose that $$ C\subseteq <!--\; \subseteq \; -->{\mathbb{R}}^n$$ is compact, and suppose $$ f:C\to <!--\; \to \; -->{\mathbb{R}}^m$$ is continuous, then the image $$ f\left(C\right)$$ is compact.

Suppose that $$ C\subseteq <!--\; \subseteq \; -->{\mathbb{R}}^n$$ is compact and $$ f:C\to <!--\; \to \; -->\mathbb{R}$$ is continuous, then there are points $$ a,b\in <!--\; \in \; -->C$$ that attain the minimum and maximum values of $$ f$$ on $$ C$$, that is, for every $$ x\in <!--\; \in \; -->C$$ we have: $\[ f\left(a\right)\le <!--\; \le \; -->f\left(x\right)\le <!--\; \le \; -->f\left(b\right) \]$