Hi,


I have asked this question on [math.stackexchange](https://math.stackexchange.com/questions/146280/on-the-set-of-divergence-to-infinity-for-sequences-of-positive-continuous-functi) but it has not received much attention, so I ask it here.


This question is partly motivated by this [one](https://mathoverflow.net/questions/34371/on-the-existence-of-a-sequence-of-positive-continuous-functions), 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 $(f\_n)$?


As noted in the comments at math.stackexchange, a necessary condition is that $S$ must be a countable interesection of $F\_{\sigma}$'s... But is this sufficient?


Thank you,
Malik



On the set of divergence to infinity for sequences of positive continuous functions
The condition that the set be an $F\_{\sigma\delta}$ is necessary and sufficient.
This is a result proved by Hahn in 1919. The reference is
H. Hahn, Ueber die Menge der Konvergenzpunkte einer Funktionenfolge,
Archiv. der Math. und Physik 28 (1919), 34-45.


