Eliminating positive-measure level sets by small Lipschitz perturbations

Sorina Barza; Martin Lind

Eliminating positive-measure level sets by small Lipschitz perturbations

Classical Analysis and ODEs 2026-07-02 v1

Authors: Sorina Barza , Martin Lind

Abstract

We establish a new regularity phenomenon of continuous functions. Specifically, given any continuous function $f$ and arbitrary $\epsilon>0$ , we construct a Lipschitz perturbation $g_\epsilon$ whose Lipschitz seminorm is less than $\epsilon$ such that every level set of $f+g_\epsilon$ has Lebesgue measure zero.

Cite

@article{arxiv.2607.02348,
  title  = {Eliminating positive-measure level sets by small Lipschitz perturbations},
  author = {Sorina Barza and Martin Lind},
  journal= {arXiv preprint arXiv:2607.02348},
  year   = {2026}
}