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On interval based generalizations of absolute continuity for functions on \(\mathbb{R}^{n}\). (English) Zbl 1388.26009

For \(\alpha \in (0,1)\), a real mapping on \(\mathbb{R}^n\) is \(\alpha\)-absolutely continuous if for all positive \(\varepsilon\) there is a positive \(\delta\) such that for any finite collection of \(\alpha\)-regular intervals \([a_i,b_i]\) the sum of lengths of these intervals being less than \(\delta\) implies that \(\sum_{i=1}^k|f(a_i - f(b_i)|< \varepsilon\). For \(\alpha =1\), the class of 1-regular intervals coincides with the class of \(n\)-dimensional cubes with sides parallel to the coordinate axes. A conjecture posed by Malý that 1-absolutely continuous functions may not be differentiable is proved together with some other properties of such functions.

MSC:

26A46 Absolutely continuous real functions in one variable
26A03 Foundations: limits and generalizations, elementary topology of the line
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