A dichotomy of sets via typical differentiability

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Colleges, School and Institutes


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.


Original languageEnglish
Article numbere41
Number of pages42
JournalForum of Mathematics, Sigma
Publication statusPublished - 4 Nov 2020


  • differentiability of Lipschitz functions, Baire category, purely unrectifiable, Banach-Mazur game