Let $f(x)=[\cos x+\sin x], 0<x<2 \pi$, where [x] denotes the greatest integer less than or equal to x. The number of points of discontinuity of f(x), is

4

We have,

$f(x)=[\cos x+\sin x]=\left[\sqrt{2} \cos \left(x-\frac{\pi}{4}\right)\right]$

We know that [x] is discontinuous at integral points. Therefore, f(x) is discontinuous at points between 0 and $2 \pi$ where $\sqrt{2} \cos \left(x-\frac{\pi}{4}\right)$ assumes integral values.

Clearly, such points are $x=\frac{\pi}{2}, \frac{\pi}{2}+\frac{\pi}{4}, \pi+\frac{\pi}{4}, \frac{3 \pi}{2}+\frac{\pi}{4}$