In category theory we say a functor is continuous if it preserves limits.
At first glance this seems like a totally obvious analogue of the notion of a continuous mapping in topology. However, at second glance it looks like a play on words, because the category-theoretic notion of "limit of a diagram" doesn't have a lot to do with the topological notion of "limit of a sequence".
Can one regard continuity of maps as a special case of continuity of functors? That is, if I have topological spaces $X, Y$ and an arbitrary mapping $f: X \to Y$, is there a natural way to cook up a functor $F_f$ between some categories such that $F_f$ is a continuous functor iff $f$ is a continuous map?