Question

Let $f\left(x\right)=[4+3\cos x],$ $x\in$$(-\frac{\pi }{2},\frac{\pi }{2}),$ where $\left[x\right]=$ greatest integer less than or equal to $x$. The number of points of discontinuity of $f\left(x\right)$ is

• A. $2$
• B. $3$
• C. $5$
• D. none of these

Answer 1

The correct option is C $5$

$[x+k]=\left[x\right]+k$ if $f$ belongs to integers.
$[4+3\cos x]=4+\left[3\cos x\right]$
Here $\cos x$ breaks at $\cos x=1/3,2/3,1$.
Therefore, $5$ points are discontinuous since $x$ belongs to $(\frac{-\pi }{2},\frac{\pi }{2})$
so $x=0$ is one point and two values in two the positive domain for $\cos x=1/3,2/3$. same in the negative domain.
So totally five points at which $f\left(x\right)$ is discontinuous.