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:

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

  • $\begingroup$ 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?' $\endgroup$ – Kavi Rama Murthy Feb 22 at 5:27
  • $\begingroup$ @KaviRamaMurthy That should be all the conditions here plus $Lf'(x_0)=Rf'(x_0)$. $\endgroup$ – Dragon Feb 22 at 5:39
  • $\begingroup$ 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. $\endgroup$ – nicomezi Feb 22 at 5:46

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