Proving that there is no continuous function $f:\Bbb R\to\Bbb R$ satisfying $f(\Bbb Q)\subset\Bbb R-\Bbb Q$ and $f(\Bbb R-\Bbb Q) \subset\Bbb Q$.

Question

How can I prove that there is no continuous function $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(\mathbb{Q}) \subset \mathbb{R}\backslash \mathbb{Q}$ and $f(\mathbb{R}\backslash \mathbb{Q} ) \subset \mathbb{Q}$?

Answer 1

Hint: Since $\bf R$ is connected and $f$ is non constant hence$f(\bf R)$ is connected and hence uncountable.

Answer 2

Hint: $f(\Bbb Q)$ is countable.

New hint given the edit: $f(\Bbb R)$ must be connected, and hence an interval, and hence must contain uncountably many irrationals.

Answer 3

HINT: Use the fact that

Between any two rational numbers, we have infinite irrational numbers

and similarly,

Between any two irrational numbers, we have infinite rational numbers.