Question on the continuity of a function

Question

Here's a question from a real analysis textbook:

Let $f\colon \mathbb{R} \to \mathbb{R}$ be continuous taking values in $\mathbb{Z}$ or in $\mathbb{Q}$. Then show that $f$ is a constant function.

What is the author trying to say here? The domain of $f$ is clearly $\mathbb{R}$, so it can accept values from both $\mathbb{Q}$ and $\mathbb{Z}$. Are they trying to say that it is continuous on either $\mathbb{Z}$ or $\mathbb{Q}$ and discontinuous on $\mathbb{R}\setminus \mathbb{Z}$ and $\mathbb{R}\setminus \mathbb{Q}$ respectively? The question is quite vague.

Answer 1

The author is saying that $f$ is a continuous function from $\Bbb R$ to $\Bbb R$ such that $f(x)\in\Bbb Z$ (respectively, "$\in \Bbb Q$") for all $x\in \Bbb R$.

And that you must prove that, with either of these assumptions, $f$ can only be constant.

Answer 2

Are you familiar with IVP? If a continuous function assumes two values it assumes all values in-between. So if it takes only integer or rationals values it has to be a constant.