# A dichotomy of sets via typical differentiability

Research output: Contribution to journal › Article › peer-review

## Standard

**A dichotomy of sets via typical differentiability.** / Dymond, Michael; Maleva, Olga.

Research output: Contribution to journal › Article › peer-review

## Harvard

*Forum of Mathematics, Sigma*, vol. 8, e41. https://doi.org/10.1017/fms.2020.45

## APA

*Forum of Mathematics, Sigma*,

*8*, [e41]. https://doi.org/10.1017/fms.2020.45

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## Author

## Bibtex

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## RIS

TY - JOUR

T1 - A dichotomy of sets via typical differentiability

AU - Dymond, Michael

AU - Maleva, Olga

PY - 2020/11/4

Y1 - 2020/11/4

N2 - 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 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.

AB - 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 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.

KW - differentiability of Lipschitz functions

KW - Baire category

KW - purely unrectifiable

KW - Banach-Mazur game

UR - https://www.cambridge.org/core/journals/forum-of-mathematics-sigma

U2 - 10.1017/fms.2020.45

DO - 10.1017/fms.2020.45

M3 - Article

VL - 8

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

SN - 2050-5094

M1 - e41

ER -