Differentiability versus continuity: Restriction and extension theorems and monstrous examples

TL;DR

This paper explores the complex relationship between continuity and differentiability in real functions, presenting new theorems, counterexamples, and elementary constructions, while addressing classical and modern questions in analysis.

Contribution

It introduces new interpolation theorems, constructs pathological examples, and provides elementary proofs of classical results, advancing understanding of differentiability and continuity.

Findings

New $D^n$-$C^n$ interpolation theorem for perfect sets

Example of a differentiable, nowhere monotone function with a zero-derivative perfect set

Elementary construction of everywhere differentiable nowhere monotone functions

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: {\em For every $n$-times differentiable $f\colon\R\to\R$ and perfect $P\subset \R$ there is a $C^n$ function $g\colon\R\to\R$ such that $f\restriction P$ and $g\restriction P$ agree on an uncountable set} and an example of a differentiable function $F\colon\R\to\R$ (which can be nowhere monotone) and of compact perfect $\mathfrak{X}\subset\R$ 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 while, paradoxically, not globally. We also present a new short and elementary construction of {\em everywhere differentiable nowhere monotone $h\colon \R\to\R$}\/ and the proofs (not involving Lebesgue measure/integration theory) of the theorems of Jarn\'\i k and of Laczkovich. The main part of this exposition, concerning continuity and first order differentiation, is presented in an narrative that answers two classical questions: \textit{To what extend a continuous function must be differentiable?} and \textit{How strong is the assumption of differentiability of a continuous function?} In addition, we overview the results concerning higher order differentiation. This includes the Whitney extension theorem and the higher order interpolation theorems related to Ulam-Zahorski problem. Finally, we discuss the results concerning smooth functions that are independent of the standard axioms ZFC of set theory. We close with a list of currently open problems related to this subject.