We are looking for an example of a function $f:\mathbb{R}\to \mathbb{R}$ such that there exist (at least) two functions $g:\mathbb{R}\to \mathbb{R}$ and some functions $h:\mathbb{R}^2\to \mathbb{R}$ with $\min\{x,y\}<h(x,y)<\max\{x,y\}$ (if $x\neq y$) and satisfying the functional equation $$g(h(x,y))= \frac{f(y)-f(x)}{y-x}, \;\;\; x\neq y.$$
Note. If $f$ is differentiable, $f'$ is invertible and $f'^{^{-1}}$ is continuous (e.g., if $f'$ is strictly increasing and continuous), then the mean value theorem (MVT) implies that $g=f'$ is a unique solution of the above functional equation (for some $h$ with the mentioned properties). Therefore, we should look for some examples in the following two cases:
Case 1) $f$ is differentiable and $f'$ is not invertible (e.g., $f(t)=t^3$)
Case 1) $f$ is not differentiable at some points (e.g., $f(t)=|t|, sgn(t)$).
Our prefer is examples with explicit formulas.
Thanks in advance