$X$, $Y$ be metric spaces and $f: X \to Y$. If $X$ is Baire and $Y$ is separable then $f$ is continuous in a dense $G_{\delta}$ of $X$ 
Let $X$, $Y$ be metric spaces and $f: X \to Y$. Suppose $X$ is Baire and $Y$ is separable. If $f ^{− 1} (O)$ is $F_{\sigma}$ for every open $O \subset Y$, show that $f$ is continuous in a dense $G_{\delta}$ of $X$

In the Dissertationes Matematicae (Rozprawy Matematyczne) book they prove that if $X$ is a Baire space, $Y$ is a space that has a countable base and $f: X \to Y$, then there exists a dense subset $D$ such that $f |_{D}$ is continuous (Theorem 3.6). Also, in How to show that the set of points of continuity is a $G_{\delta}$ they show that the set of continuity points of a function is $G_{\delta}$. Therefore $D$ would be $G_\delta$. But I don't know how to use the hypothesis that  $f ^{− 1} (O)$ is $F_{\sigma}$ for every open $O \subset Y$. Any idea please?
 A: Let me recall some definitions. Let $X$ be a topological space (say just a metric space). A nowhere dense set is a set $N\subseteq X$ which is not dense in any open set, i.e. for any non-empty open $U\subseteq X$ there is an non-empty open subset $V\subseteq U$ with $V\cap N=\emptyset$. A set is meager if it can be covered by a countable union of nowhere dense sets.
Topological space is Baire if it satisfies the Baire category theorem, that is, the complement of every meager set is dense. A set $B\subseteq X$ has the Baire property if it can be written as $U\triangle N$, where $U$ is open and $N$ is meager.
Suppose that $f:X\rightarrow Y$ is a map, where $X$ is Baire and $Y$ is second countable (i.e. it has a countable basis of open sets) such that for every open $U\subseteq Y$, the preimage $f^{-1}(U)$ has the Baire property. Then there is a comeager (complement of meager) subset $G\subseteq X$ such that $f\upharpoonright G$ is continuous. Such $G$ moreover contains a dense $G_\delta$ set. Indeed, let $(O_n)_n$ be a countable basis of topology of $Y$. Set $B_n:=f^{-1}(O_n)$. By the assumption, $B_n=U_n\triangle M_n$, for some open $U_n$ and meager $M_n$. Set $G:=X\setminus \bigcup_n M_n$. Clearly $G$ is comeager and for every $n$, $f^{-1}(O_n)\cap G$ is relatively open in $G$.
So to answer your question it remains to check that $F_\sigma$ sets have the Baire property. As I wrote in the comment, sets with the Baire property form a $\sigma$-algebra containing all open sets, so it contains all Borel sets. For the specific case of $F_\sigma$ sets, we first check that every closed set $F\subseteq X$ has the Baire property. Let $\mathrm{int}F$ denote the interior of $F$ (can be empty of course) and let $\partial F$ denote $F\setminus \mathrm{int}F$. It is easy to check that $\partial F$ is nowhere dense, in particular meager, so $F$ has the Baire property. So let now $F=\bigcup_n F_n$ be an $F_\sigma$ set, where each $F_n$ is closed. Then $F=(\bigcup_n \mathrm{int}F_n)\cup F\setminus \bigcup_n \mathrm{int}F_n$. We have $F\setminus\bigcup_n \mathrm{int}F_n\subseteq \bigcup_n \partial F_n$, so it is meager.
