Real Analysis/Topological Continuity

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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[edit]

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