Differentiability versus continuity: Restriction and extension theorems and monstrous examples

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.