It is a theorem due to Blumberg (*New Properties of All Real Functions* - Trans. AMS (1922)) and a topological space $X$ such that every real valued function admits a dense set on which it is continuous is sometimes called a Blumberg space.

Moreover, in Bredford & Goffman, *Metric Spaces in which Blumberg's Theorem Holds*, Proc. AMS (1960) you can find a characterization of Blumberg spaces. Chapter 8 of the book *Homeomorphisms in Analysis* (mentioned by Gro-Tsen in the comments, and available [here][1]) gives examples of Baire spaces which are not Blumberg spaces.


  [1]: https://www.ams.org/books/surv/054/surv054.pdf