Lipschitz restrictions of continuous functions and a simple construction of Ulam-Zahorski \(C^1\) interpolation
\textit{S. Agronsky} et al. [Trans. Am. Math. Soc. 289, 659--677 (1985; Zbl 0601.26007)] proved the following: Theorem. For every continuous function \(f:\mathbb{R}\rightarrow \mathbb{R}\) there exists a continuously differentiable function \(g:\mathbb{R}\rightarrow \mathbb{R}\) such that the set \(f=g\) is uncountable. In this paper, an elementary proof of this theorem is given. Namely, it is done using a simple argument according to which for every continuous function \(f:\mathbb{R}\rightarrow \mathbb{R}\) its restriction to some perfect set is Lipschitz.
scientific article; zbMATH DE number 6924890