Pointwise limit of a sequence of continuous functions is discontinuous at most finitely/countably many points. 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.

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.
