The left – hand derivatives of f(x) = [x] sin ($\mathrm{\pi }$x) at x = k, k is an integer, is----

(Where [.] represents GIF)

L.H.D at x = k is
$\underset{\mathrm{h}\to 0}{lim} \frac{\mathrm{f}\left(\mathrm{k}\right)-\mathrm{f}(\mathrm{k}-\mathrm{h})}{\mathrm{h}}$  (k = integer)
$\begin{array}{l}=\underset{\mathrm{h}\to 0}{lim} \frac{\left[\mathrm{k}\right]\sin \mathrm{k\pi }-[\mathrm{k}-\mathrm{h}]\sin (\mathrm{k}-\mathrm{h})\mathrm{\pi }}{\mathrm{h}}\\ =\underset{\mathrm{h}\to 0}{lim} \frac{-(\mathrm{k}-1)\sin (\mathrm{k\pi }-\mathrm{h\pi })}{\mathrm{h}}\\ =\underset{\mathrm{h}\to 0}{lim} \frac{-(\mathrm{k}-1)(-1{)}^{\mathrm{k}-1}\sinh \mathrm{\pi }}{\mathrm{h\pi }}\times \mathrm{\pi }=\mathrm{\pi }(\mathrm{k}-1)(-1{)}^{\mathrm{k}}\end{array}$