$f\left(x\right)=\sin \left|x\right|·f\left(x\right)\text{ is not differentiable at}$

$\begin{array}{l}\text{ The function breaks at }x=0\text{ and multiples of }x.\\ \text{ Hence the function is differentiable at all other points as the function is continuous at all these points }\\ \text{ At }x=0\text{ . for }f\left(x\right)\text{ to be continous }\\ \underset{x\to 0}{\lim }f\left(0^{-}\right)=f(x=0)=\underset{x\to 0}{\lim }f\left(0^{+}\right)\\ \text{f}\left(\text{x}\right)=0\text{ at x}=0\\ \text{RHL}=\underset{x\to 0}{\lim }\sin (x+h)=\sin h>0\\ \text{ L. H. L}=\underset{\text{x}\to 0}{\lim }\sin (\text{x}-\text{h})=\sin (-\text{h})<0\\ \text{ Hence, not differentiable at x}=0\\ \text{ Similarly, f}\left(\text{x}\right)\text{ is not differentiable at all multiples of }\pi .\end{array}$