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arXiv · 1909.03487

A dichotomy of sets via typical differentiability

Michael Dymond·Olga Maleva
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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 non-differentiable.

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Michael Dymond, Olga Maleva. 2019-09-08. A dichotomy of sets via typical differentiability. https://doi.org/10.1017/fms.2020.45

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