About a theorem of Volterra related to continuous functions

If 1881 Volterra proved that if two real-valued functions defined on R are continuous respectively on two dense subsets of R, then the set of common continuity points is also dense in R. The aim of my paper is to generalize this result of Volterra for complete metric spaces, proving in addition that the set of common continuity points is not countable. Particularly, a function can’t be continuous only in rational points.