No functions continuous only at points in a countable dense set

Abstract

We give a short proof that if a function is continuous on a countable dense set, then it is continuous on an uncountable set. This is done for functions defined on nonempty complete metric spaces without isolated points, and the argument only uses that Cauchy sequences converge. We discuss how this theorem is a direct consequence of the Baire category theorem, and also discuss Volterra's theorem and the history of this problem. We give a simple example, for each complete metric space without isolated points and each countable subset, of a real-valued function that is discontinuous only on that subset.