Differentiability versus continuity: Restriction and extension theorems and monstrous examples
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- by Krzysztof C. Ciesielski and Juan B. Seoane-Sepúlveda PDF
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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.References
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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
- MR Author ID: 49415
- 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
- MR Author ID: 680972
- Email: jseoane@ucm.es
- Received by editor(s): March 5, 2018
- Published electronically: September 7, 2018
- Additional Notes: The second author was supported by grant MTM2015-65825-P
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3923344