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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 $\mathbb{Q}$?

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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