On Interval Based Generalizations of Absolute Continuity for Functions on R^sub n

We study notions of absolute continuity for functions defined on Rn similar to the notion of a-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Maly that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish some containment relations of the class 1-ACWDN which consits of all functions in 1-AC which are in the Sobolev space W1,2 loc, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.