Question

Prove that the function $f$ is given by $f\left(x\right)=|x-1|,x\in R$ is not differentiable at $x=1$

Answer 1

Here $f\left(x\right)$ can be written as
$f\left(x\right)=\{\begin{array}{c}x-1x\ge 1\\ 1-xx<1\end{array}$
$f^{{}^{\prime }}\left(x\right)$ to the left of $1$ is $1$
$f^{{}^{\prime }}\left(x\right)$ to the right of $1$ is $-1$
Clearly both of them are not equal so $f\left(x\right)$ is not differentiable at $x=1$