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Let $r\geq 0$ and $f(x)=x^r \sin \frac{1}{x}$, for $x\neq 0$ and $f(0)=0$.

a) For which values of $r$, $f(x)$ is uniformly continuous on $(0,\pi)$?

b) For which values of $r$, $f$ is differentiable at $x=0$?

My solution.

a) $f'(x)= r x^{r-1} \sin \frac{1}{x}- \frac{x^r}{x^2}\cos \frac{1}{x} $

Then \begin{align*} |f'(x)|&\leq |r x^{r-1} \sin \frac{1}{x}| +|x^{r-2}| \quad\text{(since $\cos \frac{1}{x} \leq 1$)}\\ &\leq |r x^{r-2}|+|x^{r-2}|\quad\text{(since $|\sin y| \leq |y|$ when $0 \leq y \leq 1$)}\\ &\leq (|r|+1) x^{r-2} \end{align*}

When $x< \pi$ then $x^{r-2} < \pi^{r-2}$ iff $r-2 \geq 0$ or $r \geq 2$. Thus $|f'(x)| < (r+1)\pi^{r-2}$ and $f$ has a bounded derivative on $(0,\pi)$, and hence uniformly continuous on $(0, \pi)$.

b) $\lim_{x\to 0} \frac{f(x)-f(0)}{x-0}= \lim_{x\to 0} x^{r-1}\sin \frac{1}{x} $

When $r>1$ then this limit tends to $0$ as $x\to 0$, hence $f'(0)=0$ when $r>1$.

Am I right? Thanks so much.

Robert Z
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Ross
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1 Answers1

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Your answer for a) is not correct. Note that, since $f$ is continuous in $(0,\pi)$, by the extension theorem, the uniform continuity of $f$ on $(0,\pi)$ is equivalent to show that the following limits exist and they are finite $$\lim_{x\to 0^+}f(x)\quad\text{and}\quad \lim_{x\to \pi^-}f(x).$$ Can you take it from here?

Your answer for b) is correct. The function $f$ is differentiable at $0$ if and only if the following limit exists and it is finite $$\lim_{x\to 0^+}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0^+}x^{r-1}\sin \frac{1}{x}.$$ This happens if and only if $r>1$ and the derivative at $0$ is $0$.

Robert Z
  • 145,942
  • @Ross Sorry for the mess. Is my hint for a) of any help? – Robert Z Oct 23 '17 at 17:33
  • I got the answer $r>1$ from your hint. But could you please explain why by the extension theorem, the uniform continuity of $f$ on $(0,\pi)$ is equivalent to show that those limits exist and they are finite? I read that theorem but I still do not understand. – Ross Oct 23 '17 at 17:36
  • Basically, if $f$ is u,c, in $(0,\pi)$ then for any sequence $x_n\to 0^+$, the sequence $(f(x_n))_n$ is a Cauchy sequence and therefore it is convergent. On the other hand if the limit at the end points exist, $f$ can be extended to a continuous function in $[0,\pi]$ which is compact and therefore $f$ is u.c. in $(0,\pi)$. See also https://math.stackexchange.com/questions/814402/show-for-fa-to-y-uniformly-continuous-exists-a-unique-extension-to-overlin – Robert Z Oct 23 '17 at 17:44
  • Thanks so much for your nice explanation. I have 1 more question, is it enough with any sequence $x_n \to 0^+$ or do we need to consider any sequence $x_n \to 0^-$ – Ross Oct 23 '17 at 17:58
  • @Ross We need to consider any sequence which is convergent to the endpoints. – Robert Z Oct 23 '17 at 23:12