Someone asked me this question. And he said it's an exercise from Rudin's *Real and Complex Analysis*.

Does there exist a sequence of continuous functions $f_n(x)$, such that $\lim_{n \to \infty} f_n(x)=+\infty$ iff $x \in \mathbb Q$ (or irrationals)?

On the one hand, we know that if the limit of this function exists, then we can't have both as Baire's category theorem applies. But maybe it will happen that the suplim of this sequence is infinity at other points(at which the limit doesn't exists). Because lim equals infinity is the same as inflim equals infinity, so if someone can prove that rationals can't be the (countable)intersections of (countable)unions of G delta set then it's done.

But on the other hand, I suspect that if we let $f_n(x)=\cos(\pi\cdot n!x)^n \cdot n$ is an example.