JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Continuity and Differentiability

Since $a$ is an integer, $f(x)=[a+13\sin x]=a+[13\sin x]$, so the non-differentiable points are exactly the jumps of $[13\sin x]$ on $(0,\pi )$. As $x$ runs over $(0,\pi )$, $13\sin x$ rises from $0$ to $13$ on $\left(0,\frac{\pi }{2}\right]$ and falls from $13$ back to $0$ on $\left[\frac{\pi }{2},\pi \right)$. On the rising branch it crosses each integer $1,2,\dots ,12$, and on the falling branch it crosses each of $12,11,\dots ,1$ again, giving $2\times 12=24$ jumps. At $x=\frac{\pi }{2}$ it reaches $13$ at a single point, where $[13\sin x]$ steps up to $13$ and back, contributing one more. Therefore the number of points of non-differentiability is $25$.