Lipschitz restrictions of continuous functions and a simple construction of Ulam-Zahorski \(C^1\) interpolation
type: Item
instance of: scholarly article
title: Lipschitz restrictions of continuous functions and a simple construction of Ulam-Zahorski \(C^1\) interpolation
zbMATH Open document ID: 1404.26006
author: Krzysztof Chris Ciesielski
published in: Real Analysis Exchange
publication date: 2018-08-24T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rae/1530064962
review text: \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.
reviewed by: Giorgi Oniani
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 6924890
zbMATH Keywords: differentiation of functions
zbMATH Keywords: free interpolation theorem
zbMATH Keywords: extension theorems
zbMATH Keywords: Whitney extension theorem
MaRDI profile type: Publication
OpenAlex ID: W2829139287
DOI: 10.14321/REALANALEXCH.43.2.0293
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description: scientific article; zbMATH DE number 6924890