Question

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

• A. $6$
• B. $5$
• C. $4$
• D. $3$

Answer 1

The correct option is C $4$

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

So, $f\left(x\right)=\left[\sqrt{2}\cos (x-\frac{\pi }{4})\right]$

Since $\left[x\right]$ is discontinuous at integral value of $x,$ now,

$\sqrt{2}\cos (x-\frac{\pi }{4})$ is an integer at

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