# Necessary and sufficient conditions for one sided derivatives to exist

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:

$$Lf'(x_0)=\lim\limits_{x \to x_0-} \frac{f(x)-f(x_0)}{x-x_0},\,\, Rf'(x_0)=\lim\limits_{x \to x_0+} \frac{f(x)-f(x_0)}{x-x_0}$$

• Have you considered the questions: 'what are necessary conditions for $f'(x_0)$ to exist?,'what are sufficient conditions for $f'(x_0)$ to exist?' – Kavi Rama Murthy Feb 22 at 5:27
• @KaviRamaMurthy That should be all the conditions here plus $Lf'(x_0)=Rf'(x_0)$. – Dragon Feb 22 at 5:39
• The function must indeed be continuous at $x_0$ for both $Lf'$ and $Rf'$ to exist. If not, $f(x)-f(x_0)$ will not converge to $0$ at least on one side. As far as I know, there is no other necessary and sufficient condition apart from : $Rf'$ and $Lf'$ both exist. – nicomezi Feb 22 at 5:46