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A dichotomy of sets via typical differentiability

Part of: Classical measure theory Real functions Functions of several variables Set theory

Published online by Cambridge University Press:  04 November 2020

Michael Dymond  and
Olga Maleva
Michael Dymond
Affiliation:
Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria; E-mail: [email protected]
Olga Maleva
Affiliation:
School of Mathematics, University of Birmingham, Birmingham, B15 2TTUnited Kingdom; E-mail: [email protected]

Abstract

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We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function: namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has a zero-length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: in any such coverable set, a typical Lipschitz function is everywhere severely non-differentiable.

Keywords

differentiability of Lipschitz functionsBaire categorypurely unrectifiableBanach-Mazur game

MSC classification

Primary: 26B05: Continuity and differentiation questions 26A16: Lipschitz (Hölder) classes
Secondary: 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets 26A21: Classification of real functions; Baire classification of sets and functions 03E15: Descriptive set theory 28A75: Length, area, volume, other geometric measure theory
Type
Analysis
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e41
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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