$\begin{aligned} f(x)= & {[x] \sin (\pi x) } \\ \text { LHD } & =\lim _{h \rightarrow 0} \frac{f(k-h)-f(k)}{-h} \\ & =\lim _{h \rightarrow 0} \frac{[k-h] \sin \pi(k-h)-[k] \sin k \pi}{-h} \\ & =\lim _{h \rightarrow 0} \frac{(k-1) \sin (k \pi-\pi h)-k \sin k \pi}{-h} \\ & =\lim _{h \rightarrow 0} \frac{(-1)^{k+1}(k-1) \sinh \pi-0}{-h} \quad \ldots[\because k \in I] \\ & =(-1)^k(k-1) \pi\end{aligned}$