Question

The number of point(s) of discontinuity of function $f\left(x\right)=\left[\frac{6x}{\pi }\right]\cos \left[\frac{3x}{\pi }\right]$ in the interval $(\frac{\pi }{10},\frac{11\pi }{10})$ is
(where [.] is greatest integer function)

Answer 1

$f\left(x\right)=\left[\frac{6x}{\pi }\right]\cos \left[\frac{3x}{\pi }\right]$
will be discontinuous when $\frac{6x}{\pi }\text{or}\frac{3x}{\pi }\in Z$
$\because x\in (\frac{\pi }{10},\frac{11\pi }{10})$
$\Rightarrow \frac{6x}{\pi }\in (\frac{6}{10},\frac{66}{10})$ and $\frac{3x}{\pi }\in (\frac{3}{10},\frac{33}{10})$
$\Rightarrow$ Possible integral values of $\frac{6x}{\pi }$ is $1,2,3,4,5,6$ and $\frac{3x}{\pi }$ is $1,2,3.$

$\Rightarrow x\in \{\frac{\pi }{6},\frac{\pi }{3},\frac{\pi }{2},\frac{2\pi }{3},\frac{5\pi }{6},\pi \}$
$\therefore$ Number of points of discontinuity are $6.$