Uniform continuity of $\sin(x\sin x )$ for all $x\in (0,\infty)$

Question

I want to prove or disprove that $f(x)=\sin(x\sin x)$ for all $x\in (0,\infty)$ is uniformly continuous.

We know that $g(x)=x\sin x$ is not uniformly continuous on $(0,\infty)$. But what about $f$? I am not getting any hint. Any help is appreciated.

Answer 1

Hint: Consider $f(2\pi n + \pi/n) - f(2\pi n).$