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

$3\le 3+2\cos x\le 5\text{ for }x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

$f\left(x\right)=[3+2\cos x]\text{ is discontinuous at those points where }3+2\cos x\text{ is an integer }$

$\therefore 3+2\cos x=3\underset{\_}{\underset{\_}{\text{ if }}}\cos x=0\text{ so }x=\frac{-\pi }{2},\frac{\pi }{2}\text{ (not possible) }$

$3+2\cos x=4,\text{ if }\cos x=\frac{1}{2}\therefore x=\frac{\pi }{3}\text{ and }\frac{-\pi }{3}$

$3+2\cos x=5,\text{ if }\cos x=1\text{ so }x=0$

$\therefore \text{ The value of }x=3$