  We study notions of absolute continuity for functions defined on
$\mathbb{R}^n$similar to the notion of $\alpha$-absolute continuity in the
sense of Bongiorno. We confirm a conjecture of Mal\'y 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 $1-AC_{\rm WDN}$ 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.
