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.