Function whose discontinuity points are a prefixed $F_\sigma$ set in $\mathbb{R}$. I have been reading Carothers' book on real analysis and I found the following question on page 130: If E is an $F_\sigma$ set in $\mathbb{R}$, is $E=D(f)$ for some $f:\mathbb{R}\rightarrow \mathbb{R}$ ? Here $D(f)$ denotes the set of discontinuities for the function $f$. How do I solve it?

Hint: If $E = \bigcup_{n} E_n$ where $E_n$ are closed, start by finding a function $f_n: {\mathbb R} \to \\{0,1\\}$ such that $f_n(x) = 0$ for $x \
otin E_n$, while for every $x \in E$ there are points $y$ arbitrarily close to $x$ with $f_n(x) \
e f_n(y)$. Then consider $\sum_n 3^{-n} f_n$.