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The left-hand derivative of $f(x)=[x] \sin (\pi x)$ at $x=k$ where k is an integer and $[\mathrm{x}]$ is the greatest integer function, is
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$(-1)^{\mathrm{k}}(\mathrm{k}-1) \pi$
$\mathrm{f}(\mathrm{x})=[\mathrm{x}] \sin (\pi \mathrm{x})$ at $\mathrm{x}=\mathrm{k}$.
Left hand derivative, $\lim _{h \rightarrow 0} \frac{\mathrm{f}(\mathrm{k})-\mathrm{f}(\mathrm{k}-\mathrm{h})}{\mathrm{h}}$ (k-integer)
$=\lim _{h \rightarrow 0} \frac{[k] \sin k \pi-[k-h] \sin (k-h) \pi}{h}$
$=\lim _{h \rightarrow 0} \frac{-(k-1) \sin (k-h) \pi}{h}$
$\sin k \pi=0$ and $\sin (k \pi-\theta)=(-1)^{k-1} \sin \theta .$
$=\lim _{h \rightarrow 0} \frac{-(k-1)-(1)^{k-1} \sin \pi}{h \pi} \times \pi$
$=\lim _{h \rightarrow 0} \frac{-(k-1)(-1)^{k-1} \sin \pi}{h \pi} \times \pi$
$=\pi(k-1)(-1)^{k}$
Left hand derivative, $\lim _{h \rightarrow 0} \frac{\mathrm{f}(\mathrm{k})-\mathrm{f}(\mathrm{k}-\mathrm{h})}{\mathrm{h}}$ (k-integer)
$=\lim _{h \rightarrow 0} \frac{[k] \sin k \pi-[k-h] \sin (k-h) \pi}{h}$
$=\lim _{h \rightarrow 0} \frac{-(k-1) \sin (k-h) \pi}{h}$
$\sin k \pi=0$ and $\sin (k \pi-\theta)=(-1)^{k-1} \sin \theta .$
$=\lim _{h \rightarrow 0} \frac{-(k-1)-(1)^{k-1} \sin \pi}{h \pi} \times \pi$
$=\lim _{h \rightarrow 0} \frac{-(k-1)(-1)^{k-1} \sin \pi}{h \pi} \times \pi$
$=\pi(k-1)(-1)^{k}$
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