Lipschitz functions with unexpectedly large sets of nondifferentiability points

Marianna Csörnyei, David Preiss and Jaroslav Tišer

Abstract and Applied Analysis, 2005, vol. 2005, 1-13

Abstract:

It is known that every G δ subset E of the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function on ℝ 2 has a point of differentiability in E . Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct a G δ set E ⊂ ℝ 2 containing a dense set of lines for which there is a pair of real-valued Lipschitz functions on ℝ 2 having no common point of differentiability in E , and there is a real-valued Lipschitz function on ℝ 2 whose set of points of differentiability in E is uniformly purely unrectifiable.

Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:451030

DOI: 10.1155/AAA.2005.361

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