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Abstract
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 nondifferentiable.
Original language  English 

Article number  e41 
Number of pages  42 
Journal  Forum of Mathematics, Sigma 
Volume  8 
DOIs  
Publication status  Published  4 Nov 2020 
Bibliographical note
Publisher Copyright:© The Author(s), 2020. Published by Cambridge University Press.
Keywords
 differentiability of Lipschitz functions
 Baire category
 purely unrectifiable
 BanachMazur game
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Dive into the research topics of 'A dichotomy of sets via typical differentiability'. Together they form a unique fingerprint.Projects
 1 Finished

Differentiability and Small sets
Engineering & Physical Science Research Council
1/07/16 → 30/06/19
Project: Research Councils