Lipschitz functions with unexpectedly large sets of nondifferentiability points

Abstract

It is known that every $G_{\delta }$ 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 $G_{\delta }$ set $E\subset {ℝ}^2$ 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.