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Let $\alpha\in\mathbb{R}$ and $\beta >0$ consider the function:

$$f_{\alpha,\beta}(x)=f(x)= \left\{ \begin{array}{lcc} x^\alpha\sin(x^\beta) & if & 0 < x \leq 1 \\ \\ 0& if & x =0 \\ \end{array} \right.$$

Prove $f$ is Lipchitz in $[0,1]$

My attempt:

Let $x,y\in [0,1]$ and $\alpha + \beta >0$ then

Note that $f(x)-f(y)=(x^\alpha-y^\alpha)\sin x^\beta+y^\alpha(\sin x^\beta-\sin y^\beta)$

$$\dfrac{|f(x)-f(y)|}{|x-y|}=\dfrac{|(x^\alpha-y^\alpha)\sin x^\beta+y^\alpha(\sin x^\beta-\sin y^\beta)|}{|x-y|}\leq\dfrac{|(x^\alpha-y^\alpha)||\sin x^\beta|+|y^\alpha||\sin x^\beta-\sin y^\beta|}{|x-y|}$$

$$\leq \dfrac{|x^\alpha-y^\alpha|+2|y^\alpha|}{|x-y|}$$

Here, I'm stuck. Can someone help me?

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  • $\begingroup$ I think the question should say for what values of $\alpha$ and $ \beta$ is the function Lipschitz? $\endgroup$ Apr 1, 2020 at 23:14
  • $\begingroup$ If $\alpha + \beta < 0$ does it not make the function singular at origin? $\endgroup$
    – r9m
    Apr 1, 2020 at 23:14
  • $\begingroup$ You should write the condition on $\alpha,\beta$ at the beginning. $\endgroup$
    – zhw.
    Apr 2, 2020 at 4:14

1 Answer 1

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If you are allowed to prove the derivative is bounded:

$f'(x)=\alpha x^{\alpha-1}\sin(x^\beta)+\beta x^{\alpha+\beta-1}\cos(x^\beta)$. Therefore, \begin{align} |f'(x)|&\leq |\alpha x^{\alpha-1}\sin(x^\beta)| + |\beta x^{\alpha+\beta-1}\cos(x^\beta)| \\ &\leq |\alpha|+|\beta| \end{align} under the right assumptions on $\alpha$ and $\beta$. From my inequality, sufficient conditions should be clear. Given that $\beta>0$, you can also show that they are necessary.

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  • $\begingroup$ If $\alpha + \beta>0$ how can you guarantee that $|\alpha x^{\alpha-1}\sin(x^\beta)| + |\beta x^{\alpha+\beta-1}\cos(x^\beta)| \\ \leq |\alpha|+|\beta|$ Thanks for answer. $\endgroup$
    – rcoder
    Apr 1, 2020 at 23:16
  • $\begingroup$ You need more. Hint: $\lim_{x\to 0} \frac{\sin(x)}{x}=1$ How does changing the powers cause blow up? That's for the first term in the derivative. For the second, it's even easier since $\cos(0^\beta)=1$ so blow up is determined by the $x^{\alpha+\beta-1}$ term. $\endgroup$
    – ProfOak
    Apr 1, 2020 at 23:21

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