On Interval Based Generalizations of Absolute Continuity for Functions on R_n

Abstract

We study notions of absolute continuity for functions defined on R n similar to the notion of α-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý that 1absolutely 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 W 1,2 loc , are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.