These are the properties that such functions should (could?) have:
1) $f(\mathbb R)=\mathbb R$
2) $f$ is everywhere discontinuous
3) $\mathbb Q \subseteq f( \mathbb I)$
4) $f(\mathbb Q) \subset \mathbb I$
That is, such functions should (could?) have all these properties simultaneously: They map the whole $\mathbb R$ onto the whole $\mathbb R$, they are everywhere discontinuous, they map the set of all irrational numbers $\mathbb I$ into some set that contains all rationals, and they map the set of all rationals into some subset of the set of all irrationals.
Remark: This is not a homework, I am just eager into doing research of everywhere discontinuous functions.