TITLE: Pointwise limit of a sequence of continuous functions is discontinuous at most finitely/countably many points.
QUESTION [5 upvotes]: Let $\{f_{n}\}$ be a sequence of functions in $C[0,1]$ such that $f_{n}\to f$ pointwise. Then $f$ has at most finitely (or probably countably) many discontinuities. Is this statement TRUE or FALSE? Any kind of help will be appreciated.

REPLY [4 votes]: By Egorov's theorem, pointwise convergence almost everywhere implies uniform convergence on a set of measure arbitrarily close to $1$. Therefore $f$ is continuous except possibly on a set of measure $0$.
Edit: Disregard this, it's wrong. I apologize. See Aram's link below in comments, it answers the question. The link is Give an example of a sequence of continuous functions which converges on a compact set to a function that has an infinite number of discontinuities., from which we can see both that the above is wrong (by choosing C to be a nowhere dense set of positive measure, having the sequence converge to the characteristic function of C) and that the answer to your question is false, by noting that a set of positive measure is uncountable.