Question

Let a function $f\left(x\right)$ be discontinuous at $n$ number of points in $R$ and is continuous at $x=0.$ Then which of the following is/are true?

• A. The minimum number of points of discontinuity of $|f\left(x\right)|$ will be $0.$
• B. The maximum number of points of discontinuity of $f\left(|x|\right)$ will be $2n.$
• C. The maximum number of points of discontinuity of $|f\left(x\right)|$ will be $n.$
• D. $f\left(|x|\right)$ will be discontinuous at $2n$ number of points.

Answer 1

Answer: (A), (B), (C)

The maximum number of points of discontinuity of $|f\left(x\right)|$ will be $n.$

We know, $|f\left(x\right)|$ is obtained by keeping the portion of $f\left(x\right)$, where $f\left(x\right)\ge 0$ and reflection of portion of $f\left(x\right)$ in $x-\text{axis}$ where $f\left(x\right)<0$
$\Rightarrow$ If $f\left(x\right)$ is discontinuous at $x={\alpha }_i,i=1,2,\cdots ,n,$ then $|f\left(x\right)|$ will be discontinuous at maximum ${\alpha }_i,i=1,2,\cdots ,n$
$\Rightarrow$ Maximum number of points of discontinuity is $n.$
But there may be at some points, where $|f\left(x\right)|$ will become continuous as shown in figure


Example:
$f\left(x\right)=\{\begin{array}{cc}{e}^x,& 2n<x<2n+1\\ -e^x,& 2n+1\le x<2n+2\end{array}$
where $n\in R^{+}$

Now,
$f\left(|x|\right)$ is obtained by removing the portion of $f\left(x\right)$ when $x<0$ and by making reflection of portion when $x\ge 0$ in $y-\text{axis}.$
$\Rightarrow$ If $f\left(x\right)$ is discontinuous at $x={\alpha }_i,i=1,2,\cdots ,n,$ then $f\left(|x|\right)$ will be discontinuous at $\pm {\alpha }_j,j=1,2,\cdots ,r$ where ${\alpha }_j>0$ and $r\le n$
(as points of discontinuity, where $x<0$ i.e. ${\alpha }_i<0$ will be removed)

$\Rightarrow$ Maximum number of points of discontinuity of $f\left(|x|\right)$ will be $2n.$