Problem: Prove that for any function f from a countable subset M of $\mathbb{R}$ there exists a sequence of continuous functions {f_n} that converges pointwise to f on M.
Context: This was put forward in lecture as a way to deduce that in the Baire-Osgood Theorem the condition of completeness was needed to get at least one point of continuity.
I have been struggling on a way to make headway on this problem. We are using Carothers Real Analysis and we have studied up to Chapter 12, this includes Baire Category Theorem, Arzela-Ascoli Theorem, and Stone-Wierstrass.
I feel that I am missing something something obvious that would simplify the problem significantly but I do not see how I can use any of the results we have found in class or in the text.
I would appreciate any insight or suggestions on how to go about the proof.