The question is simple. I state it for $\mathbb{R}^3$, although I am pretty sure that dimension 3 should not be relevant here.
It is well known that the divergence of a vector field at a point is defined as a limit ratio involving the flux (see https://en.wikipedia.org/wiki/Divergence#Definition) if it exists. Besides, if $F$ is a $\textbf{continuously differentiable}$ vector field $F=(F_1,F_2,F_3)$, the divergence exists and it is given by $$F = \frac{\partial F_{1}}{\partial x}+\frac{\partial F_{2}}{\partial y}+\frac{\partial F_{3}}{\partial z}. \quad \quad (*)$$
Assume now that $F$ is differentiable (but not necessarily continuously differentiable). My question is:
Does the divergence of $F$ (limit ratio involving the flux) always exist?
Provided that the divergence exists, can be always computed via (*) or does it exist a counterexample?
In general, books (or at least the ones that I have checked) just state the result for the continuously differentiable case, but do not tell anything about what happens when only differentiability is required.