Mathematics - Continuity - Differentiability Question with Solution | TestHub

Clearly ${\varphi }^{{}^{\prime }}(x)=2x\cdot \sin \left(\frac{1}{x}\right)-\cos \left(\frac{1}{x}\right)$ if $x\ne 0$

$$=0\text{ if }x=0.$$

$\Rightarrow {\varphi }^{{}^{\prime }}(x)$ is distinuous at $x=0$, as $\cos \left(\frac{1}{x}\right)$ is oscillating in the neighbour hood of ' $\underline{0}$ '