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arXiv · 1306.4291

On interval based generalizations of absolute continuity for functions on $\mathbb{R}^n$

Michael Dymond·Beata Randrianantoanina·Huaqiang Xu
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Abstract

We study notions of absolute continuity for functions defined on $\mathbb{R}^n$similar 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 $1-AC_{\rm WDN}$ 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.

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Michael Dymond, Beata Randrianantoanina, Huaqiang Xu. 2014-03-31. On interval based generalizations of absolute continuity for functions on $\mathbb{R}^n$. https://arxiv.org/abs/1306.4291

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