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.