Question

The number of points where $f\left(x\right)=[\sin x+\cos x]$ (where []denotes greatest integer function), $x$ $ϵ(0,2\pi )$ is not continuous is :

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is C $5$

$\sin x+\cos x=\sqrt{2}(\cos \frac{\pi }{4}\sin x+\sin \frac{\pi }{4}\cos x)=\sqrt{2}\sin (x+\frac{\pi }{4})$

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

We know, $\left[x\right]$ is not continuous where $x$ is integer

Hence f will not be continuous where $\sqrt{2}\sin (x+\frac{\pi }{4})$ will be an integer

Hence points of discontinuities in the given interval are,

$x=\frac{\pi }{2},\frac{3\pi }{4},\pi ,\frac{3\pi }{2},\frac{7\pi }{4}$