Showing uniform continuity of $x^m \sin \left ( \frac{1}{x^n}\right )$

Question

For $m, n \in \mathbb{N} \setminus \{0 \}$,

$f(x) = \begin{cases} x^m \sin \left ( \frac{1}{x^n}\right ) & x \neq 0\\ 0 & x = 0 \end{cases}$

where the domain of $f$ is $[-1, 1]$.

$f(x) \to 0$ as $x \to 0^+$ or $x \to 0^-$. So the function is continuous.

I am struggling to show whether the function is or is not uniformly continuous.

Answer 1

If a function is continuous on any $E\subseteq \mathbb{R}$, then it is uniformly continuous for any compact (closed and bounded) subset of $E$.

Where do you have continuity of $f$? Can you construct a closed and bounded subset of that domain where $f$ is continuous on to argue that $f$ must be uniformly continuous on that interval?