On interval based generalizations of absolute continuity for functions on Rⁿ

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We study notions of absolute continuity for functions defined on Rⁿsimilar to the notion of α-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý 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 containment relations of the class 1-AC WDN which consits of all functions in 1-AC which are in the Sobolev space W¹,²loc, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.