Let $f:(a,b) \to \mathbb{R}$ be a differentiable function. For simplicity assume that $0 \in (a,b)$. We know by the mean value theorem that, for each $x$ there exists $c_x \in (0, x)$ (or $c_x \in (x,0)$) with $$ \frac{f(x)-f(0)}{x} = f^{'}(c_x).$$ Consider $S_x$ the set of all values $c_x$ for which the mean value theorem holds. Now define $g: (a,b) \to \mathbb{R}$ with $g(x) = \inf{S_x}$.
Define $g(0)$ and provide conditions on $f$ such that
1- $g$ is continuous on $\mathbb{R}$.
2 - $g$ is differentiable on $\mathbb{R}$.
For example, if $f^{'}$ is one-to-one and continuous (differentiable) then $g$ is also continuous (differentiable). Because in that case $g(x) = (f^{'})^{-1}(\frac{f(x)-f(0)}{x}).$
More interesting cases are when $f^{'}$ is not one-to-one. Any ideas about the general behavior of g?