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2018 Lipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski \(C^1\) Interpolation
Krzysztof Chris Ciesielski
Real Anal. Exchange 43(2): 293-300 (2018). DOI: 10.14321/realanalexch.43.2.0293

Abstract

We present a simple argument showing that for every continuous function \(f\colon\mathbb{R}\to\mathbb{R}\), its restriction to some perfect set is Lipschitz. We will use this result to provide an elementary proof of the \(C^1\) free interpolation theorem, that for every continuous function \(f\colon\mathbb{R}\to\mathbb{R}\) there exists a continuously differentiable function \(g\colon\mathbb{R}\to\mathbb{R}\) which agrees with \(f\) on an uncountable set. The key novelty of our presentation is that no part of it, including the cited results, requires from the reader any prior familiarity with Lebesgue measure theory.

Citation

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Krzysztof Chris Ciesielski. "Lipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski \(C^1\) Interpolation." Real Anal. Exchange 43 (2) 293 - 300, 2018. https://doi.org/10.14321/realanalexch.43.2.0293

Information

Published: 2018
First available in Project Euclid: 27 June 2018

zbMATH: 06924890
MathSciNet: MR3942579
Digital Object Identifier: 10.14321/realanalexch.43.2.0293

Subjects:
Primary: 26A24
Secondary: 26B05

Rights: Copyright © 2018 Michigan State University Press

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Vol.43 • No. 2 • 2018
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