Lipschitz functions with unexpectedly large sets of nondifferentiability points

Marianna Csörnyei, David Preiss and Jaroslav Tišer

Abstract and Applied Analysis, 2005, vol. 2005, issue 4, 361-373

Abstract: It is known that every Gδ 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 ℝ2 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 Gδ set E ⊂ ℝ2 containing a dense set of lines for which there is a pair of real‐valued Lipschitz functions on ℝ2 having no common point of differentiability in E, and there is a real‐valued Lipschitz function on ℝ2 whose set of points of differentiability in E is uniformly purely unrectifiable.

Date: 2005
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https://doi.org/10.1155/AAA.2005.361

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2005:y:2005:i:4:p:361-373

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