$F_{\sigma}$ subsets of $\mathbb{R}$ Suppose $C \subset \mathbb{R}$ is of type $F_{\sigma}$. That is $C$ can be written as the union of $F_{n}$'s where each $F_{n}$'s are closed. Then can we prove that each point of $C$ is a point of discontinuity for some $f: \mathbb{R} \to \mathbb{R}$. I refered this link on wiki : < and in the follow up subsection they given this result. I would like somebody to explain it more precisely.

I believe you're looking for something like the construction mentioned here.