Cantor sets of low density and Lipschitz functions on $C^1$ curves

Abstract

We characterize the functions f : [0, 1] −→ [0, 1] for which there exists a measurable set C ⊆ [0, 1] of positive measure satisfyingAs an application, we prove that on any C 1 curve it is possible to construct a Lipschitz function that cannot be approximated by Lipschitz functions attaining their Lipschitz constant.