No function is continuous only at points in a countable dense subset

Abstract

We give a short proof, that can be used in an introductory real analysis course, that if a function that is defined on the set of real numbers is continuous on a countable dense set, then it is continuous on an uncountable set. This is done for functions defined on complete metric spaces without isolated points, and the argument only uses that Cauchy sequences converge, and we prove the version related to Volterra's theorem. 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 conclude with a simple example, for each complete metric space without isolated points and each set that is a countable union of closed subsets, of a real-valued function that is discontinuous only on that set.