Absolutely continuous functions and differentiability in \({\mathbb R}^n\) (Q1856814)

It is known that the definition of the notion of absolute continuity of functions in a finite-dimensional space may depend on the set used to define it. In this note the geometrical definition due to \textit{J. Malý} [J. Math. Anal. Appl. 231, 492-508 (1999; Zbl 0924.26008)] is relativized to subsets of the finite-dimensional space using a class of sets containing those from \textit{M. Csőrnyei} [J. Math. Anal. Appl. 252, 147-166 (2000; Zbl 0981.26009)] and it is shown that each of these concepts of absolute continuity can be used to characterize function differentiable almost everywhere. Moreover, it is noted that the notion of ``absolute continuity up to a set of small measure'' does not depend on the shape of the set. All the results hold, with the same proof, for functions with values in a Banach space with the Radon-Nikodým property.