Geodesic
A dichotomy of sets via typical differentiability
Forum of Mathematics, Sigma, Tome 8 (2020)

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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. 
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     author = {Michael Dymond and Olga Maleva},
     title = {A dichotomy of sets via typical differentiability},
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Michael Dymond; Olga Maleva. A dichotomy of sets via typical differentiability. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.45

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