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Show that the function $f(x)=\sin \frac1x$ is neither continuous nor uniformly continuous on $[0,\infty)$

Show that the function $f(x)=\sin \frac1x$ is neither continuous nor uniformly continuous on $[0,\infty)$

Continuity:
Let $f(x)=\sin \frac1x$, we show that $f(x)$ is not continuous on $[0,\infty)$
$f$ is continuous on any interval $(0,r)$ where $0\leq r<\infty$ because $\lim_{x\to 0}f(x) = 0$ and also as $\lim_{x\to 0}f(x)= \sin \frac 1 0$ is $0$
$\lim_{x\to 0^+}f(x) = \sin \frac 1 0$ as $0< \sin \frac 1 0 < 0$
Uniform Continuity:
Let $f(x)=\sin \frac1x$, we show that $f(x)$ is not uniformly continuous on $[0,\infty)$
As $f$ is uniformly continuous on any interval \$(a,

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