Need help proving $h$ is continuous at $0$

Question

Here's the problem:

Define $h: \mathbb{R} \to \mathbb{R}$ by

Prove $h$ is continuous at $0$.

Usually, I have a good idea how to prove this when the cases are $x=0$ and $x\neq0$, but I'm a bit thrown off here since it says $x$ is rational/irrational. Can someone help explain this problem to me?

I think I'm supposed to use $\epsilon-\delta$ proof, but I'm not sure.

Answer 1

Hint:

$$\left|x\sin\frac1x\right|\le|x|\to 0$$

Answer 2

You could equivalently use sequential continuity: show that for all $a_n\to 0$, we have $h(a_n)\to h(0)=0$. Using the fact that $|\sin|\leq 1$, this shouldn't be too hard.