Question

Let $f\left(x\right)=[3+2\cos x],x\in (-\frac{\pi }{2},\frac{\pi }{2})$ where $[.]$ denotes the greatest integer function. The number of point(s) of discontinuity of $f\left(x\right)$ is

Answer 1

$3<3+2\cos x\le 5$ for $x\in (-\frac{\pi }{2},\frac{\pi }{2})$
$f\left(x\right)=[3+2\cos x]$ is discontinuous at those points where $3+2\cos x$ is an integer.
Hence,
$3+2\cos x=4,$ if $\cos x=\frac{1}{2}.$
So, $x$ have two values $\frac{\pi }{3}$ and $-\frac{\pi }{3}$
$3+2\cos x=5,$ if $\cos x=1.$
So, $x=0$
$\therefore$ The number of values of $x=2+1=3$