If $f\left(x\right)=\left\{\begin{array}{ccc}x{e}^{-\left(\frac{1}{\left|x\right|}+\frac{1}{x}\right)};& \mathrm{i}\mathrm{f}& x\ne 0\\ 0;& \mathrm{i}\mathrm{f}& x=0\end{array}\right.$ then which of the following is correct?

Prove that the function $f\left(x\right)=\left\{\begin{array}{ll}{x}^2-1& ; x\ge 1\\ 1-x& ; x<1\end{array}\right.$ is not differentiable at $x=1$

Prove that function $f\left(x\right)=\left|x\right|$ is not differentiable at $x=0$.