If $f(\mathbb{R})$ is a subset of $\mathbb{Q}$, is it continuous or differentiable I'm doing a practice paper before my exam and I came across the following question, but I don't have the memo. I know if a function is differentiable then it it must continuous, so that rules out choice C. Other than that, I don't know how to answer this. I'm assuming it's continuous the answer is D but I'm really uncertain

Let $f:\mathbb{R}\to\mathbb{R}$ such that $f(\mathbb{R})\subset\mathbb{Q}$. Then:

A. $f$ is not continuous.

B. $f$ is continuous but not differentiable.

C. $f$ is differentiable but not continuous.

D. $f$ is differentiable and monotonic.

E. $f$ is continuous if and only if $f$ is constant.
A. $f$ is not continuous since for conrinuous functions $f(\Bbb R)$ is an interval, but $\Bbb Q$ is not the interval.

D. Yes, if $f$ is a constant function. For nonconstant ones this is false.

E. True.
A. $f$ is not continuous since for conrinuous functions $f(\Bbb R)$ is an interval, but $\Bbb Q$ is not the interval.

D. Yes, if $f$ is a constant function. For nonconstant ones this is false.

E. True.