Extreme non-differentiability of typical Lipschitz mappings

We show that no matter what subset of a normed space is given, a typical 1-Lipschitz mapping into a Banach space is non-differentiable at a typical point of the set in a very strong sense: the set of partial limits of its derivative ratios, which is separable, contains all linear operators of norm at most 1 from any fixed separable subspace of the operator space.

For subsets of finite-dimensional normed spaces which can be covered by a countable union of closed purely 1-unrectifiable sets this extreme non-differentiability holds for a typical Lipschitz mapping at every point.

Both results are new even for Lipschitz mappings with a finite-dimensional codomain.