  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.
