Question

The number of points, where $f(x)=[\sin x+\cos x]$ (where [.] denotes the greatest integer function) and $x\in (0,2\pi )$ is not continuous, is

• A. 3
• B. 4
• C. 5
• D. 6

Answer 1

Answer: (C)

Given, $f(x)=[\sin x+\cos x]$
$=\left[\sqrt{2}\left(\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x\right)\right]$
$=\left[\sqrt{2}\sin \left(x+\frac{\pi }{4}\right)\right]$
We know that, greatest integer function is discontinuous on integer values.
Function $\sqrt{2}\sin \left(x+\frac{\pi }{4}\right)$ will gives integer values at
$x=90^{\circ },135^{\circ },180^{\circ },270^{\circ },315^{\circ }$
Hence, there are five points in the given interval, in which $f(x)$ is not continuous.