Is there a function $f:\mathbb{R} \to \mathbb{R}$ such that a set of all continuous points of $f$ is "given" $G_\delta$ set?

Question

I'm a student studying analysis. There is a famous statement that for all $f:\mathbb{R} \to \mathbb{R}$, the set $C(f):=\{x \in \mathbb{R}:f \text{ is continuous at } x\}$ is $G_\delta$, and I'm considering a (some kind of) converse of this statement.

For every $A \in G_\delta$, is there a function $f:\mathbb{R} \to \mathbb{R}$ such that $C(f)=A$?

For open $A \subseteq \mathbb{R}$, define $$ f= \chi_{\mathbb{Q} \setminus A}+\chi_{\mathbb{R} \setminus A} $$ where $\chi$ is a characteristic function, then we have $C(f)=A$.

For closed $A \subseteq \mathbb{R}$, define $$ f(x)=d(x, A) \cdot (\chi_{\mathbb{Q}}(x)-\chi_{\mathbb{R} \setminus \mathbb{Q}}(x)) $$ where $d(x,A)$ is distance between $x$ and $A$, then $C(f)=A$ holds.

If $A=\mathbb{R} \setminus \mathbb{Q}$, then the Thomae's function works.

However, I don't know how can I construct such $f$ for general $G_\delta$ sets (or a counterexample). Can I have any good idea?

Thanks.

Under PROOFS OF 2' in this 20 December 2006 sci.math post I give about 20 references (books and papers) for this result. 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, May 25 '20 at 16:45
See Set of continuity points of a real function AND Is every $G_\delta$ set the set of continuity points of some function $f$? — Dave L. Renfro, May 25 '20 at 16:51

Is there a function $f:\mathbb{R} \to \mathbb{R}$ such that a set of all continuous points of $f$ is "given" $G_\delta$ set?

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