Question

The function $f\left(x\right)$ is defined as $|\left[x\right]x|$ for $-1<x\le 2$.
For what values of $x$ is $f\left(x\right)$ discontinuous ?

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

Answer: (B), (C)

$2$

$f=|\left[x\right]x|=|\left[x\right]||x|$

$f\left(x\right)=\{\begin{array}{cc}-x,& -1<x<0\\ 0,& 0\le x<1\\ x,& 1\le x<2\\ 4,& x=2\end{array}$

$\text{L.H.L}=\underset{x\to 1^{-}}{lim}f\left(x\right)=0$
$\text{R.H.L}=\underset{x\to 1^{+}}{lim}f\left(x\right)=1$
$\text{L.H.L}\ne \text{R.H.L}$

$\text{L.H.L}=\underset{x\to 2^{-}}{lim}f\left(x\right)=2$
$\text{R.H.L}=\underset{x\to 2^{+}}{lim}f\left(x\right)=4$
$\text{L.H.L}\ne \text{R.H.L}$

$f\left(x\right)$ discontinuous at $x=1,2$