Real Analysis Exchange

Lipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski \(C^1\) Interpolation

Krzysztof Chris Ciesielski

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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.

Article information

Source
Real Anal. Exchange, Volume 43, Number 2 (2018), 293-300.

Dates
First available in Project Euclid: 27 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1530064962

Digital Object Identifier
doi:10.14321/realanalexch.43.2.0293

Mathematical Reviews number (MathSciNet)
MR3942579

Zentralblatt MATH identifier
06924890

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
Secondary: 26B05: Continuity and differentiation questions

Keywords
differentiation of partial functions extension theorems Whitney extension theorem

Citation

Ciesielski, Krzysztof Chris. Lipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski \(C^1\) Interpolation. Real Anal. Exchange 43 (2018), no. 2, 293--300. doi:10.14321/realanalexch.43.2.0293. https://projecteuclid.org/euclid.rae/1530064962


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