Let T be any map of the set X to itself. A natural question to ask is whether there is a topology on X with respect to which T is continuous. Clearly the discrete and indiscrete topologies make any map continuous, so a better question is `When is there a nice topology on X with respect to which T is continuous?'

We consider the case when `nice' is taken to mean compact Hausdorff, and characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant.