Decomposing Euclidean space with a small number of smooth sets

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.

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“…(Compare also [35].) Note, that an earlier, weaker version of the theorem was proved by Juris Steprāns [108].…”

Section: 3mentioning

confidence: 96%

Differentiability versus continuity: Restriction and extension theorems and monstrous examples

Ciesielski

Seoane‐Sepúlveda

2018

Bull. Amer. Math. Soc.

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“…Klee [22] proved that no separable Banach space can be covered by fewer than 2 ℵ0 hyperplanes. Steprāns [28] proved the consistency of covering R n+1 by fewer than continuum smooth manifolds of dimension n.…”

Section: Covering a Square By Functionsmentioning

confidence: 99%

Continuous Ramsey Theory on Polish Spaces and Covering the Plane by Functions

2004

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We investigate the Ramsey theory of continuous pair-colorings on complete, separable metric spaces, and apply the results to the problem of covering a plane by functions.The homogeneity number hm(c) of a pair-coloring c : [X] 2 → 2 is the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2 ω , c min and cmax, which satisfy hm(c min ) ≤ hm(cmax) and prove: Theorem. (1) For every Polish space X and every continuous pair-coloring c : [X] 2 → 2 with hm(c) > ℵ 0 , hm(c) = hm(c min ) or hm(c) = hm(cmax).(2) There is a model of set theory in which hm(c min ) = ℵ 1 and hm(cmax) = Diagram 1 in the Introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Diagram 1 can be separated.

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“…It is consistent with ZFC thatdec(B 1 , C) < d.There are also some interesting results concerning the value of dec(C, D 1 ), where D 1 is the class of all (partial) differentiable functions. It has been proved by Morayne (see Steprāns[149, Thm 6.1]) that…”

mentioning

confidence: 99%