Let $f:\mathbb{R} \to \mathbb{R}$ be a function and $x_0 \in \mathbb{R}$. We denote the left and right hand derivative respectively by $Lf'$ and $Rf'$. If the two exist and match at a point, we say that $f$ is differentiable at that point. I have the following queries:
$(1)$ What are the necessary conditions for $Lf'(x_0)$ and $Rf'(x_0)$ to exist? My guess is that we need continuity. What else?
$(2)$ What are the sufficient conditions for $Lf'(x_0)$ and $Rf'(x_0)$ to exist? Continuity is not enough (for example, $f(x)=x^{1/3}$ at $x=0$). What else do we need?
In case of any confusion, I'm defining the one sided derivatives:
$$Lf'(x_0)=\lim\limits_{x \to x_0-} \frac{f(x)-f(x_0)}{x-x_0},\,\, Rf'(x_0)=\lim\limits_{x \to x_0+} \frac{f(x)-f(x_0)}{x-x_0}$$