at x=0 Left hand limit and Right hand limit are equal , for derivability we check LHD(left hand derivative) and RHD(Right hand derivative) , those are not equal so function at x=0 is not differentiable.
$LHD $ = $f'(a^{-})$ = $\frac{f(a+h)-f(a)}{h}$ where $h->0^{-}$
$RHD$ = $f'(a^{+})$ = $\frac{f(a+h)-f(a)}{h}$ where $h->0^{+}$
LHD = $f'(0^{-})$ = $\frac {f(0+h)-f(0)}{h}$ = $\frac {\left | 0+h \right | -\left | 0 \right |}{h}$ = $\lim_{h->0^{-}} \frac{\left | h \right |}{h} = \frac{-h}{h}=-1$
RHD = $f'(0^{+})$ = $\frac {f(0+h)-f(0)}{h}$=$\frac {\left | 0+h \right | -\left | 0 \right |}{h}$ = $\lim_{h->0^{+}} \frac{\left | h \right |}{h} = \frac{h}{h}=1$
$LHD\neq RHD$