Rationals can be the set of continuity of a function?

Question

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 — Alessandro Codenotti, May 17 '19 at 20:43

score 1 · Accepted Answer · answered May 17 '19 at 20:47

1

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

answered May 17 '19 at 20:47

1 Answers1