The function $f(x)=|| x|-1|, x \in R$ is differentiable at all $x \in R$ except at the points

1, 0, -1

We have,

$f(x)=|| x|-1|= \begin{cases}|x-1|, & x \geq 0 \\ |x+1|, & x<0\end{cases}$

$\Rightarrow f(x)= \begin{cases}x-1 & , ~x \geq 1 \\ 1-x &, ~0 \leq x<1 \\ x+1 &, ~-1 \leq x \leq 0 \\ -x-1 & ,~ x<-1\end{cases}$

It is evident from the curve y = f(x) that the function f(x) is not differentiable at x = -1, 0, 1.