Question

The number of point(s) where $f\left(x\right)=[\sin x+\cos x]$ $($ where $[.]$ denotes greatest integral function $),x\in (0,2\pi )$ is not continuous

Answer 1

$f\left(x\right)=[\sin x+\cos x]$ will be discontinuous iff $\sin x+\cos x\in Z$
We know that range of $\sin x+\cos x$ is $[-\sqrt{2},\sqrt{2}].$ So, possible integral values of $\sin x+\cos x=-1,0,1$

$\left(i\right)\sin x+\cos x=-1\Rightarrow \sin (\frac{\pi }{4}+x)=-\frac{1}{\sqrt{2}}\Rightarrow x=\pi ,\frac{3\pi }{2}\left(ii\right)\sin x+\cos x=0\Rightarrow \tan x=-1\Rightarrow x=\frac{3\pi }{4},\frac{7\pi }{4}\left(iii\right)\sin x+\cos x=1\Rightarrow \sin (\frac{\pi }{4}+x)=\frac{1}{\sqrt{2}}\Rightarrow x=\frac{\pi }{2}$

$\therefore$ Function is discontinuous at
$x\in$ $\{\frac{\pi }{2},\frac{3\pi }{4},\pi ,\frac{3\pi }{2},\frac{7\pi }{4}\}$

Thus, the number of points at which $f\left(x\right)$ is discontinuous is $5.$