$f:[a,b] \to \mathbb R$ continuous a.e. , is there a sequence of functions of finite discontinuity on $[a,b]$ converging uniformly to $f$ on $[a,b]$?

Question

Let $f:[a,b] \to \mathbb R$ be a function continuous almost everywhere , then does there exist a sequence of functions $\{f_n\}$ on $[a,b]$ , where each $f_n$ has at most finitely many discontinuity point , such that $\{f_n\}$ converges uniformly to $f$ on $[a,b]$ ?

Answer 1

If $f_n \colon [a,b] \rightarrow \mathbb{R}$ is a sequence of functions that are continuous at $x_0 \in [a,b]$ and $f_n$ converge uniformly to $f \colon [a,b] \rightarrow \mathbb{R}$ then $f$ is also continuous at $x_0$. Thus, if the set of discontinuities of $f$ is not countable, you can't approach $f$ uniformly by a sequence of functions with finitely many discontinuity points.