Derivative of x sin of reciprocal

Let $f:\mathbb{R}\to\mathbb{R}$ be the function defined by $$f(x)=\begin{cases} \lim_{h\to 0} \frac{(x+h)\sin\left(\frac{1}{x+h}\right)-x\sin\frac{1}{x}}{h}, & x\neq 0 \ 0, & x=0. \end{cases}$$ Then which one of the following statements is NOT true?