Limits (Calculus)

B), C)

First of all note that f(x) is continuous at x=0.
Further, we have
f'(0)=\lim_{\Delta x\to0}\dfrac{f(0+\Delta x)-f(0)}{\Delta x}=\lim_{\Delta x\to0} \Delta x\sin\left(\dfrac{1}{\Delta x}\right)=0
Hence, f is differentiable at x=0.
Next, for x≠0,
f'(x)=2x\sin\left(\dfrac{1}{x}\right)-\cos\left(\dfrac{1}{x}\right)
which has no limit as x→0.
Hence f'(x) is not continuous at x=0.

Agreed, Thankyou sir.