Does there exists a function $f:\Bbb{R}\to \Bbb{R}$ s.t $f$ is continuous on a dense $G_{\delta}$ set and discontinuous a.e on $\Bbb{R}$? Does there exists any  function $f:\Bbb{R}\to \Bbb{R}$ such that $f$ is continuous on a dense $G_{\delta}$ subset of $\Bbb{R}$ and discontinuous almost everywhere on $\Bbb{R}$ ?
Attempt : (Yes)
We know the real line $\Bbb{R}$ can be decomposed into disjoint union of two small sets i.e $\Bbb{R}=N\cup M$ where $N$ is Lebesgue null and $M$ is meager (see here).
Since $\Bbb{R}$ is second category (non-meager) and $M$ is meager implies $N$ is of second category. Since complement of a first category set contains a dense $G_{\delta}$ subset of $\Bbb{R}$ , let $G\subset N$ a dense $G_{\delta}$ subset of $\Bbb{R}$.
According to this
How to construct a real valued function which is continuous only on a given $G_\delta$ subset of $\mathbb{R}$? , we can construct a  function which is continuous  only on $G$ .
As $N$ is Lebesgue null and $G\subset N$ implies $m(G) =0$ . Hence $f$ is discontinuous everywhere.
Is my approach is correct ?
Edit: I can produce an explicit $G_{\delta}$ set of measure $0$.

The first part - existence of null dense $G_\delta$ set - can be show explicitly: let $q_n$ be enumeration of rationals, let $U_k = \bigcup_n (q_n - 2^{-n-k}, q_n + 2^{-n-k})$, then $\cap_k U_k$ is null (as $\mu(U_k) \leq 2^{-k}$), dense (as it contains $\mathbb Q$) and $G_\delta$ (as each $U_k$ is open).
I don't see any construction simpler than general to find a function continuous exactly on this set, but if we substitute our $U_k$ in general construction, we get pretty much explicit function.
