Compute $f'(0)$ and check if $f'$ is continuous or not 
Given the function 
  $$f(x)=\begin{cases}x^{4/3}\sin(1/x)&\text{if $x\neq 0$}\\0&\text{if $x=0$}\end{cases}.$$
  
  
*
  
*Compute  $f'(0)$;
  
*Is $f'$ continuous on  $\mathbb{R}$.

I am unsure of how to solve this but will post what I have.
Computing the part 1 it shows that  $f'(0) =0$. (unless I computed it wrong)
For part two I am totally lost on how to prove this, but am thinking that since $f'=0$ wouldn't that make the function not continuous?. Any help is appreciated!

Similar problems like this one are around on the Internet.
For the first one you can do Squeeze theorem since the sine modulates between $-1$ and $1$. Second part: For the derivative you need to use the limit definition $\lim\limits_{x\to0}\frac{f(x)-f(0)}{x-0}$, using your answer from part a) for $f(0)$ and since the exponent is more than $1$, you will find that $f'(0)$ exists (although the derivative is NOT continuous) 
