Any union of a countable number of closed sets is a set of discontinuity points of some function

Question

I am following Calculus lectures on YT (in Russian) and stumbled upon the following statement that was left without proof --

Any union of a countable number of closed sets is a set of discontinuity points of some function

I would appreciate it if someone can show how a strategy for proving this statement.

This statement comes right after the lecturer shows that number of discontinuity points of a monotone function is at most countable.

my translation might be off, apologies for that. I am trying to say that number of sets in the union is countable. This seems to be a weaker version of the theorem above https://math.stackexchange.com/questions/3003058/any-countable-set-of-real-numbers-is-set-of-discontinuities-of-some-monotone-fun (unless I missed your point) — Rotkiv, Dec 01 '20 at 16:28
Okay, I thought it might be a confusion with Froda's theorem, that the number of discontinuity points of a monotone function is at most countable. — hardmath, Dec 01 '20 at 16:31
It's fairly well known that any $F_{\sigma}$ set can be a discontinuity set for some function. See Set of continuity points of a real function and Is every $G_\delta$ set the set of continuity points of some function $f$? Under PROOFS OF 2' in this 20 December 2006 sci.math post I give about 20 references (books and papers) for this result. (continued) — Dave L. Renfro, Dec 01 '20 at 16:37
FYI, this was first proved for functions $f:{\mathbb R} \rightarrow {\mathbb R}$ by William H. Young in 1903 and Henri Lebesgue in 1904 (independently), extended to functions $f:{\mathbb R}^n \rightarrow {\mathbb R}$ by William H. Young in 1905. I believe the first metric space version appears on pp. 193-194 of Hans Hahn's 1932 book. — Dave L. Renfro, Dec 01 '20 at 16:37
@hardmath: Regarding Froda's theorem, see B. S. Thomson's answer to Proof of Froda's theorem. — Dave L. Renfro, Dec 01 '20 at 16:44

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