Let $f:\mathbb R^m\to\mathbb R^n$ and $g:\mathbb R^n\to\mathbb R^p$ be two functions that each possess a complete Jacobian matrix of partial derivatives $J_f$ and $J_g$ respectively, which are well-defined everywhere. Is it true that the composite $h=g\circ f:\mathbb R^m\to\mathbb R^p$ also has a complete well-defined Jacobian matrix, and that $J_h(x)=J_g(f(x)) J_f(x)$ for each $x\in\mathbb R^m$?
I think that the answer is no. This looks like the chain rule, except that there is no assumption that $f$ and $g$ are differentiable. Having well-defined partial derivatives everywhere does not imply differentiability (classic counterexample is $f(x,y)=\frac{xy}{x^2+y^2}$, $f(0,0)=0$). All proofs of the chain rule hinge on the assumption of differentiability.
However, I am struggling to come up with a counterexample to the claim.