I was wondering, is every homeomorphism between open subsets of $\mathbb{R}^n$ necessarily (component-wise) ACL, or at least "partial-differentiable along coordinate directions" (or just along a given direction) almost everywhere?
Note: We recall absolute continuity implies differentiability almost-everywhere (cf. e.g. Wikipedia) for single-variable functions; consequently ACL implies partial-differentiability (along a given, or finitely-many given, directions) ae.
Remark: Differentiability ae should apply in $n=1$. Proof: A homeomorphism $\mathbb{R}\rightarrow \mathbb{R}$ is necessarily monotone, hence by Lebesgue's theorem it is differentiable ae.