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Let \( f(x)=\left\{\begin{array}{cl}(x-1) \sin \frac{1}{(x-1)}, & \text { if } x \neq 1 \\ 0 & , \text { if } x=1\end{array}\right. \). Then, which of the following is true?
(A) \( f \) is differentiable at \( x=1 \) but not at \( x=0 \)
(B) \( f \) is neither differentiable at \( x=0 \) nor at \( x=1 \)
(C) \( f \) is differentiable at \( x=0 \) and \( x=1 \)
(D) \( f \) is differentiable at \( x=0 \) but not \( x=1 \)
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