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

We have, f(x) $=$|x$-$1|
$f(1) = |1 - 1| = 0$

$Rf'(1) = \mathop {\lim }\limits_{h \to 0} \cfrac{{f(1 + h) - f(1)}}{h} = \mathop {\lim }\limits_{h \to 0} \cfrac{{|1 + h - 1| - 0}}{h}$

$= \mathop {\lim }\limits_{h \to 0} \cfrac{{|h|}}{h} = \mathop {\lim }\limits_{h \to 0} \cfrac{h}{h} = 1$

and $Lf'(1) = \mathop {\lim }\limits_{h \to 0} \cfrac{{f(1 - h) - f(1)}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \cfrac{{|1 - h - 1| - 0}}{{ - h}}$

$= \mathop {\lim }\limits_{h \to 0} \cfrac{{| - h|}}{h} = \mathop {\lim }\limits_{h \to 0} \cfrac{h}{{ - h}} = - 1$

Thus,$Rf'(1) \ne Lf'(1)$

This shows that f(x) is not differentiable at $x = 1.$