Testing if a function is uniformly continuous on an open interval

Question

Let $f(x)=\sqrt[3]{4x}\sin\left(\frac{5}{x^2}\right)$, is this uniformly continuous on $(0,2)$?

I started out using the Weierstrass M-Test but quickly realized that this does not apply since I don't have a sequence of functions, which the definitions I read require. Most of the definitions and rules I see only apply to a sequence of functions.

How else can I test this to see if it is uniformly continuous.

Edit

I just realized the Weierstrass M-Test is a test for uniform convergence, not continuity.

Answer 1

Hints: use the fact a function continuous on a compact interval is continuous.

Now, clearly $(0,2)$ is not compact. But $[0,2]$ is; can $f$ be extended by continuity into a continuous function $\tilde{f}\colon[0,2]\mapsto \mathbb{R}$?

And if so, can you derive uniform continuity of $f$ from the uniform continuity of this $\tilde{f}$?

More details (spoilers):

There is no issue at $2$, since $f$ clearly has a limit there. Now, can you show that $f$ admits a limit $\ell$ at $0$? (using, for instance, that $\sin$ is bounded and $x^{1/3}$ converges to $0$...) There, set $\tilde{f}(0)=\ell$, and Bob's your uncle.