In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality automatically implies that $f$ is $\mathcal C^1$ (hence real-analytic, see below)?

By complex analysis, we know the answer is positive when $n=2$.

In higher dimensions, Liouville's theorem characterizes conformal maps as Möbius transformations, but it is stated for $f \in W^{1,n}$ in Wikipedia. Is it known whether it also holds when $f$ is assumed everywhere differentiable?