Abstract and Applied Analysis
Volume 2005 (2005), Issue 4, Pages 361-373
doi:10.1155/AAA.2005.361
Lipschitz functions with unexpectedly large sets of nondifferentiability points
1Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
2Department of Mathematics, Faculty of Electrical Engineering, Technical University of Prague, Prague 166 27, Czech Republic
Received 12 January 2004
Copyright © 2005 Marianna Csörnyei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.