The left-hand derivative of $\mathrm{f}(x)=[x] \sin (\pi x)$, at $x=\mathrm{k}, \mathrm{k}$ is an integer and $[\cdot]$ is the greatest integer function, is

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Aicharya · Accepted Answer

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

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

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