Question

If $[.]$ denotes the greatest integer function and $x\in (0,20),$ then the number of points of discontinuity of $f\left(x\right)=[2x+[x\left]\right]$ is

• A. $19$
• B. $20$
• C. $39$
• D. $40$

Answer 1

The correct option is C $39$

$f\left(x\right)=[2x+[x\left]\right]=\left[2x\right]+\left[x\right]$
We know that $\left[x\right]$ is discontinuous when $x\in Z$
If $n\in Z$, then
$\underset{x\to {\left(\frac{2n-1}{2}\right)}^{-}}{lim}f\left(x\right)=2n-2+n-1=3n-3$
$\underset{x\to {\left(\frac{2n-1}{2}\right)}^{+}}{lim}f\left(x\right)=2n-1+n-1=3n-2$
$\therefore f\left(x\right)$ is not continuous at $x=\frac{2n-1}{2}$
$x=\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots ,\frac{39}{2}$
$20$ numbers are there.

$\underset{x\to n^{-}}{lim}f\left(x\right)=2n-1+n-1=3n-2$
$\underset{x\to n^{+}}{lim}f\left(x\right)=2n+n=3n$
$\therefore f\left(x\right)$ is not continuous at $x=n$
$x=1,2,3,\dots ,19$
(These are $19$ numbers)
Hence, the total number of points of discontinuity is $20+19=39$