Real Analysis Exchange

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

Krzysztof Chris Ciesielski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 27 June 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

differentiation of partial functions extension theorems Whitney extension theorem


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.

Export citation