Let $f(x) = \left\{ \begin{array} {1 1} (x-1) \sin \bigg(\large\frac{ 1 }{x-1}\bigg),\normalsize & \quad\text{ if x \(\neq\) 1 }\\ 0 ,& \quad \text{if x =1}\\ \end{array} \right. $.Then which one of the following is true?

Question

( A ) f is neither differentiable at x = 0 nor at x = 1
( B ) f is differentiable at x = 1 but not at x = 0
( C ) f is differentiable at x = 0 but not at x = 1
( D ) f is differentiable at x = 0 and at x = 1