Real Analysis/Topological Continuity

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

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.