This question is motivated by this one on MathOverflow, which contains an example of a sequence $(f_n)$ of positive continuous functions on $\mathbb{R}$ such that $f_n(x) \rightarrow \infty$ if and only if $x \in \mathbb{Q}$.
My question is the following :
For a given sequence of positive continuous functions $(f_n)$ on $\mathbb{R}$, denote by $S((f_n))$ the set of divergence to $\infty$ : $S((f_n)):=\{ x \in \mathbb{R} : f_n(x) \rightarrow \infty \}.$
Is there a necessary and sufficient condition for a given set $S$ to be $S((f_n))$ for some sequence $(f_n)$?
Note that $S((f_n)) = \bigcap_{N} \bigcup_{k} \bigcap_{n \geq k} \{x: f_n(x)>N\},$ so a necessary condition is that $S$ must be a countable intersection of countable unions of $G_{\delta}'s$...