Question

The function $f\left(x\right)$ is defined as $|\left[x\right]x|$ for $-1<x\le 2$. The number of points where $f\left(x\right)$ is non differentiable is

Answer 1

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

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

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

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

$f\left(x\right)$ discontinuous at $x=1,2$
Clearly, $f\left(x\right)$ is not differentiable at $x=1,2$ and also ​​​​​​​$f\left(x\right)$ is not differentiable at $x=0$

Hence, the number of points where $f\left(x\right)$ is non differentiable is $3$.