## Real Analysis Exchange

### Lipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski $C^1$ Interpolation

Krzysztof Chris Ciesielski

#### 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

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

#### 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