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We give a short proof, that can be used in an introductory real analysiscourse, that if a function that is defined on the set of real numbers iscontinuous on a countable dense set, then it is continuous on an uncountableset. This is done for functions defined on complete metric spaces withoutisolated points, and the argument only uses that Cauchy sequences converge, andwe prove the version related to Volterra's theorem. We discuss how this theoremis a direct consequence of the Baire category theorem, and also discussVolterra's theorem and the history of this problem. We conclude with a simpleexample, for each complete metric space without isolated points and each setthat is a countable union of closed subsets, of a real-valued function that isdiscontinuous only on that set.

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