Let $(a,b)$ be an open interval in $\mathbb{R}$.
$f:(a,b)\rightarrow\mathbb{R}$
$z\in (a,b)$ and let $f$ be continuous in $z$.
Let $f$ be differentiable over $(a,b)\setminus\{z\}$.
My question is whether the following assertion is true or false:
If the left side limit of $f'$ exists in $z$, then the left side derivative of $f$ exists in $z$. In other words if
$\lim\limits_{x\rightarrow z-0}f'(x)$ exists, then
$f'_{-}(z)$ exists and $f'_{-}(z) = \lim\limits_{x\rightarrow z-0}f'(x)$
