$\sin(x)\cos(\pi/x)$ is not uniformly continuous?

Question

Is this function defined on $(0,\infty)$ uniformly continuous? $$\sin(x)\cos(\pi/x)$$

Efforts:

I know $\cos(\pi/x)$ is not uniformly continuous on the domain. Can I use this to solve the problem?

Any hints?

Answer 1

This function is uniformly continuous. Note that $|f(x)| \leq |\sin x| \to 0$ as $x \to 0$ and $|f(x)-\sin x| \leq |1-\cos \frac {\pi} x| \to 0$ as $x \to \infty$ . Combined with the facts that $\sin x$ is uniformly continuous and any continuous function is uniformly continuous on $[-N,N]$ for each $N$ you can show that $f$ is uniformly continuous.