Most of the functions that I have seen have their discontinuities on rationals and continuities on irrationals!
I am wondering if there is any exampe of some function whose continuities are rationals? Or is other words
The set of continuities of a function $f:\mathbb{R}\to\mathbb{R}$ can be $\mathbb{Q}$?
No, the set of points of continuity of a function $\Bbb R\to\Bbb R$ is always a $G_\delta$ set and $\Bbb Q$ isn't.
For the first claim see this question.
For the second claim see this question