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

We present a simple argument that for every continuous function f: ℝ → 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: ℝ → ℝ there exists a continuously differentia...

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Bibliographic Details
Published inReal analysis exchange Vol. 43; no. 2; pp. 293 - 300
Main Author Ciesielski, Krzysztof Chris
Format Journal Article
LanguageEnglish
Published Michigan State University Press 01.01.2018
Subjects
Continuous functions
Differentiable functions
Interpolation
Lebesgue measures
Mathematical functions
Mathematical monotonicity
Mathematical theorems
Open intervals
Uncountable sets
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ISSN0147-1937
1930-1219

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Summary:We present a simple argument that for every continuous function f: ℝ → 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: ℝ → ℝ there exists a continuously differentiable function g: ℝ → 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 the Lebesgue measure theory. Mathematical Reviews subject classification: Primary: 26A24; Secondary: 26B05 Key words: differentiation of partial functions, extension theorems, Whitney extension theorem
ISSN:0147-1937
1930-1219