Let $f\left(x\right) = \begin{cases} 0, & x < 0 \\ x^2, & x \geq 0 \end{cases}$ $\;\;\;$ Check the differentiability of $f\left(x\right)$ at $x=0$.
$\begin{aligned}
\text{Left Hand Derivative} = Lf'\left(0\right) & = \lim\limits_{h \to 0} \; \dfrac{f\left(0-h\right)-f\left(0\right)}{-h} \\
& = \lim\limits_{h \to 0} \; \dfrac{0-0}{-h} \\
& = 0
\end{aligned}$
$\begin{aligned}
\text{Right Hand Derivative} = Rf'\left(0\right) & = \lim\limits_{h \to 0} \; \dfrac{f\left(0+h\right)-f\left(0\right)}{h} \\
& = \lim\limits_{h \to 0} \; \dfrac{\left(0+h\right)^2-0}{h} \\
& = \lim\limits_{h \to 0} \; h \\
& = 0
\end{aligned}$
Since $Lf'\left(0\right) = Rf'\left(0\right)$, $f\left(x\right)$ is differentiable at $x=0$.