Question

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

• A. 6
• B. 5
• C. 4
• D. 3

Answer 1

The correct option is B

5

$[cosx+sinx]=\left[\sqrt{2}cos\right(x-\frac{\pi }{4}\left)\right]$
We know that [x] is discontinuous at integral values of x,
Now, $\sqrt{2}cos(x-\frac{\pi }{4})$ is an integer.
$\Rightarrow x=\frac{\pi }{2},\frac{3\pi }{4},\pi ,\frac{3\pi }{2},\frac{7\pi }{4}$.

The function is discontinuous at 5 points in its domain