Eliminating positive-measure level sets by small Lipschitz perturbations

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.