Differentiability versus continuity: Restriction and extension theorems and monstrous examples

• Published in: Bulletin (new series) of the American Mathematical Society 2019-04, Vol.56 (2), p.211-260
• Main Authors: Ciesielski, Krzysztof C., Seoane-Sepúlveda, Juan B.
• Format: Article
• Language: English
• Subjects: Axioms; Continuity; Continuity (mathematics); Differentiation; Interpolation; Real variables; Set theory; Theorems
• ISSN: 0273-0979; 1088-9485
• DOI: 10.1090/bull/1635

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.