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Check if the function $f(x) = x\ln x +{\frac{1}{x}}\sin(x^3)$ is uniformly continuous is the intervals: $(i) (0,20);(ii)(0,∞)$

I think it's not, but I can't figure out counter examples.

Any ideas?

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    HINT: what is the value of the function near zero? Evaluate the limit when $x\to 0^+$, and the limit when $x\to 20$ or $x\to \infty$ to take an idea of the bahavior of the function. Then you can think about if it is uniformly continuous or not. – 2017-02-05
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    I've checked the function for $x=0$ and it doesn't diverge, so I think it's uniformly continuos in $(0,20)$. Not sure for the case $ii$ – 2017-02-05

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(i) $f(x)$ converges as $x\to0,20$ and is continuous in-between so we can extend $f$ to $[a,b]$ by continuity. The resulting function $\overline{f}$ is continuous on a compact set and thus u.c. by Heine-Cantor.

(ii) Hint: Show that for any $M>0$ you can always find a subinterval of $(0,\infty)$ in which $\lvert f'(x)\rvert>M$. By the mean value theorem, this tells you $f$ is not u.c. there.