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Differentiability versus continuity: Restriction and extension theorems and monstrous examples


Authors: Krzysztof C. Ciesielski and Juan B. Seoane-Sepúlveda
Journal: Bull. Amer. Math. Soc. 56 (2019), 211-260
MSC (2010): Primary 26A24, 54C30, 46T20, 58B10, 54A35, 26A21, 26A27, 26A30, 54C20, 41A05
DOI: https://doi.org/10.1090/bull/1635
Published electronically: September 7, 2018
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Abstract: The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between continuous and differentiable real valued functions of one real variable. The truly new results include, among others, the $ D^n$-$ C^n$ interpolation theorem: For every $ n$-times differentiable $ f\colon \mathbb{R}\to \mathbb{R}$ and perfect $ P\subset \mathbb{R}$, there is a $ C^n$ function $ g\colon \mathbb{R}\to \mathbb{R}$ such that $ f\restriction P$ and $ g\restriction P$ agree on an uncountable set and an example of a differentiable function $ F\colon \mathbb{R}\to \mathbb{R}$ (which can be nowhere monotone) and of compact perfect $ \mathfrak{X}\subset \mathbb{R}$ such that $ F'(x)=0$ for all $ x\in \mathfrak{X}$ while $ F[\mathfrak{X}]=\mathfrak{X}$. Thus, the map $ \mathfrak{f}=F\restriction \mathfrak{X}$ is shrinking at every point though, paradoxically, not globally. However, the novelty is even more prominent in the newly discovered simplified presentations of several older results, including a new short and elementary construction of everywhere differentiable nowhere monotone $ h\colon \mathbb{R}\to \mathbb{R}$ and the proofs (not involving Lebesgue measure/integration theory) of the theorems of Jarník--Every differentiable map $ f\colon P\to \mathbb{R}$, with $ P\subset \mathbb{R}$ perfect, admits differentiable extension $ F\colon \mathbb{R}\to \mathbb{R}$--and of Laczkovich--For every continuous $ g\colon \mathbb{R}\to \mathbb{R}$ there exists a perfect $ P\subset \mathbb{R}$ such that $ g\restriction P$ is differentiable. The main part of this exposition, concerning continuity and first-order differentiation, is presented in a narrative that answers two classical questions: To what extent must a continuous function be differentiable? and How strong is the assumption of differentiability of a continuous function? In addition, we give an overview of the results concerning higher-order differentiation. This includes the Whitney extension theorem and the higher-order interpolation theorems related to the Ulam-Zahorski problem. Finally, we discuss the results concerning smooth functions that are independent of the standard axioms of ZFC set theory. We close with a list of currently open problems related to this subject.


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Additional Information

Krzysztof C. Ciesielski
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310—and—Department of Radiology, MIPG, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6021
Email: KCies@math.wvu.edu

Juan B. Seoane-Sepúlveda
Affiliation: Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis y Matemática Aplicada, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: jseoane@ucm.es

DOI: https://doi.org/10.1090/bull/1635
Keywords: Continuous function, differentiable function, points of continuity, points of differentiability, Whitney extension theorem, interpolation theorems, independence results, classification of real functions, Baire classification functions, Pompeiu derivative
Received by editor(s): March 5, 2018
Published electronically: September 7, 2018
Additional Notes: The second author was supported by grant MTM2015-65825-P
Article copyright: © Copyright 2018 American Mathematical Society