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2017 On Interval Based Generalizations of Absolute Continuity for Functions on \(\mathbb{R}^{n}\)
Michael Dymond, Beata Randrianantoanina, Huaqiang Xu
Real Anal. Exchange 42(1): 49-78 (2017).

Abstract

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 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 some containment relations of the class $1\textit{-} AC_{\rm WDN}$ which consits of all functions in $1\textit{-}AC$ 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.

Citation

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Michael Dymond. Beata Randrianantoanina. Huaqiang Xu. "On Interval Based Generalizations of Absolute Continuity for Functions on \(\mathbb{R}^{n}\)." Real Anal. Exchange 42 (1) 49 - 78, 2017.

Information

Published: 2017
First available in Project Euclid: 27 March 2017

zbMATH: 06848941
MathSciNet: MR3702556

Subjects:
Primary: 26A03, 26A04
Secondary: 26A05

Rights: Copyright © 2017 Michigan State University Press

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Vol.42 • No. 1 • 2017
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