Several properties of continuity on sets of real numbers can be extended by examining continuity from a Topological standpoint. In topology, an alternate definition (i.e. other than the standard "epsilon-delta" real analysis definition) is usually used. This definition applies to any function between sets, not just to metric spaces.

Definition Let ${\displaystyle A\subseteq \mathbb {R} }$. Also, let ${\displaystyle f:A\to \mathbb {R} }$. ${\displaystyle f(x)}$ is continuous at ${\displaystyle x=c}$ iff for every open subset ${\displaystyle V}$ of ${\displaystyle f(A)}$, ${\displaystyle U\subseteq f^{-1}(V)}$ is open in ${\displaystyle A}$.

It must be mentioned here that the term "Open Set" can be defined in much more general settings than the set of reals or even metric spaces; however, for use in Real Analysis, the definition of Open Set that you are already familiar with will definitely suffice.

Theorem

For any continuous function f:A->B, U compact => f(U) compact.