Question

Number of points where $f\left(x\right)=\left|x\text{sgn}(1-x^2)\right|$ is non-differentiable is

Answer 1

$\text{sgn}(1-x^2)=\{\begin{array}{cc}1,& x\in (-1,1)\\ 0,& x\in \{-1,1\}\\ -1,& x\in (-\infty ,-1)\cup (1,\infty )\end{array}$
$\Rightarrow f\left(x\right)=\{\begin{array}{cc}\left|x\right|,& x\in (-1,1)\\ 0,& x\in \{-1,1\}\\ |-x|=|x|,& x\in (-\infty ,-1)\cup (1,\infty )\end{array}$

Hence, $f\left(x\right)$ has discontinuity at $x=-1,\text{and}1$.
Also, $f\left(x\right)$ will have a sharp corner at $x=0$.
Hence, total points of non-differentiability $=3$