Suppose that $f : \mathbb{R}\rightarrow \mathbb{R}$ is continuous and that $f(x)\in \mathbb{Q}$ for all $x \in \mathbb{R}$. Prove $f$ is constant.

Question

I am really stuck on this proving this statement, so could someone please help me get through this. Thank You.

P.S. This can be proved through elementary analysis results instead of going into topology.

Answer 1

Hint: Use the intermediate value theorem and the fact that there is an irrational in between any two distinct rationals.