So we all know of a function that's continuous on the irrationals but discontinuous on the rationals, but what about this:

Are there functions $\displaystyle f : [0, 1] \rightarrow \mathbb{R}$

continuous on $\displaystyle \mathbb{Q}$ and discontinuous on $\displaystyle \mathbb{R} \backslash \mathbb{Q}$? Justify.

I'm quite sure the answer's no, but not entirely sure on how to prove it.

Any help would be much appreciated, thanks!