Lipschitz functions with unexpectedly large sets of nondifferentiability points. (English) Zbl 1098.26010

Csörnyei, Marianna; Preiss, David; Tišer, Jaroslav

Abstr. Appl. Anal. 2005, No. 4, 361-373 (2005).

It is well known by the Rademacher theorem that a Lipschitz function \(f:\mathbb{R}^n\to\mathbb{R}^m\) is differentiable almost everywhere or conversely: the set of points where \(f\) is not differentiable is very small. For one-dimensional functions, this result is sharp in that sense that for every (Lebesgue) null set \(E\subset\mathbb{R}\) there exists a Lipschitz function which is nondifferentiable on \(E\). This sharpness can not be extended for higher-dimensional spaces. It can be shown that there exists a null set \(E\subset\mathbb{R}^2\) such that every real-valued Lipschitz function on \(\mathbb{R}^2\) is differentiable at some point of \(E\).
The authors provide a special \(G_\delta\) subset \(E\subset \mathbb{R}^2\) containing a dense set of lines for which the set of points of differentiability of particular Lipschitz functions inside of \(E\) is extremely small. More exactly, they construct
(1) two real-valued Lipschitz functions on \(\mathbb{R}^2\) with no common points of differentiability in \(E\) (or equivalently: a vector-valued Lipschitz function on \(\mathbb{R}^2\) which is nondifferentiable on \(E)\),
(2) a real-valued Lipschitz function on \(\mathbb{R}^2\) whose set of points of differentiability in \(E\) is uniformly purely unrectifiable (i.e., this set meets every rectifiable curve in a set of one-dimensional measure zero).

Reviewer: Jörg Thierfelder (Ilmenau)

MSC:

26B35	Special properties of functions of several variables, Hölder conditions, etc.
26A27	Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
49J50	Fréchet and Gateaux differentiability in optimization
26B05	Continuity and differentiation questions
46G05	Derivatives of functions in infinite-dimensional spaces
26A21	Classification of real functions; Baire classification of sets and functions
54C50	Topology of special sets defined by functions

Keywords:

differentiability; \(G_\delta\) set; points of nondifferentiability