5. continuity and differentiation — Maths STD 12 Science — Question

$f(0 + 0) = \mathop {\lim }\limits_{h \to 0} |0 + h| = 0$ and $f(0 - 0) = \mathop {\lim }\limits_{h \to 0} |0 - h| = 0$

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

$ = \mathop {\lim }\limits_{h \to 0} \frac{h}{h}\,\,(h$ being positive $)=1$

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

$ = \mathop {\lim }\limits_{h \to 0} \frac{h}{{ - h}} \,\, (h$ being positive $) = -1.$

$\therefore Rf'(0) \ne Lf'(0)$. 

The function  $f$  is not differentiable.