Components of level sets of uniform co-Lipschitz functions on the plane

Consider a co-Lipschitz uniformly continuous function $f$defined on the plane. Let $n(f)$ be the maximal number of components of its level set. In the present paper we settle a question of B. Randrianantoanina, concerning the dependence of $n(f)$ on the quantitative characteristics of the mapping. We prove that $n(f)$ is bounded from above by a simple function of the co-Lipschitz and the ``weak Lipschitz'' constants of $f$, and show that our estimate is sharp. We also prove additional properties of the level sets.