Given a locally Lipschitz function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g'(x) > 0$ for all $x \neq 0$ and $g(0) = 0$, can we conclude that it is continuously differentiable? If not, what additional condition(s) must $g$ satisfy? Physically, $g$ corresponds to the disspation term in a second order system, and I need $g'$ to be continuous to ultimately apply the Bendixson's Criterion.
I'm able to show that $g$ is also differentiable at $0$ using the properties given and the definition of the derivative, but unsure how to show $g'$ is continuous to conclude that it is $C^1$.