JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Continuity and Differentiability

The term $4|2x+3|$ has a corner at $x=-\frac{3}{2}$. The step $9\left[x+\frac{1}{2}\right]$ jumps wherever $x+\frac{1}{2}$ is an integer, that is at the half-integers; inside $(-20,20)$ these run from $-19.5$ to $19.5$, which is $40$ points, and the corner at $x=-\frac{3}{2}$ is itself one of them and so adds nothing new. The step $-12[x+20]$ jumps wherever $x$ is an integer; inside $(-20,20)$ these are $-19,\dots ,19$, which is $39$ points, none shared with the half-integers. The function is therefore non-differentiable at $40+39=79$ points.