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 : R → R and perfect P ⊂ R there is a C n function g : R → R such that f P and g P agree on an uncountable set and an example of a differentiable function F : R → R (which can be nowhere monotone) and of compact perfect X ⊂ R such that F (x) = 0 for all x ∈ X while F [X] = X; thus, the map f = F X is shrinking at every point while, 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 : R → R and the proofs (not involving Lebesgue measure/integration theory) of the theorems of Jarník: Every differentiable map f : P → R, with P ⊂ R perfect, admits differentiable extension F : R → R and of Laczkovich: For every continuous g : R → R there exists a perfect P ⊂ R such that g P is differentiable. The main part of this exposition, concerning continuity and first order differentiation, is presented in an narrative that answers two classical questions: To what extend a continuous function must be differentiable? and 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.2010 Mathematics Subject Classification.

Obtaining algebrability in subsets of real functions

“…This previous result was used to show the algebrability of several different classes of functions ( [10]), such as the set of Sierpiński-Zygmund functions (see [10] and, also, [24]), the set of bounded approximately continuous functions that are discontinuous almost everywhere ( [10], [25]), the set of nowhere Hölder functions, the set of differentiable nowhere monotone functions ([2], [10], [18]), among others.…”

[S]-linear and convex structures in function families

“…As said in the Introduction, in this section and the next one we will focus our attention on two specific families, namely, the set of hypercyclic vectors respect to a linear operator, and the class of differentiable functions that are nowhere monotone on the real line. Just as a brief summary, let us recall that the existence of these "differentiable monsters" (differentiable nowhere monotone functions) dates back to the work by Katznelson and Stromberg (1974, [29]), although several new constructions have been appearing since then (see, e.g., [18] for a recent expository work covering this class of functions).…”

Classical vs. non-Archimedean analysis: an approach via algebraic genericity

“…For instance, we can name algebrability and strong algebrability defined in [5,8], respectively. We refer the interested reader to [1,2,[4][5][6][10][11][12][13][14][14][15][16][17][19][20][21][22]24,25,[27][28][29]45] for a current state of the art on this topic.…”