2018/06/28 by Krzysztof Chris Ciesielski, Ciesielski, Krzysztof C., Juan B. Seoane Sepúlveda +1
Mathematics · #26A21 #26A24 #26A27 #26A30 #41A05 #46T20 #54A35 #54C20 #54C30 #58B10 #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1806.10994
openalex publication_date 2018/06/28 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
The aim of this expository article is to present recent developments in the\ncenturies old discussion on the interrelations between continuous and\ndifferentiable real valued functions of one real variable. The truly new\nresults include, among others, the Dn-Cn interpolation theorem: em For\nevery n-times differentiable f colon R\→ R and perfect P\⊂ R there\nis a Cn function g colon R\→ R such that f restriction P and\ng restriction P agree on an uncountable set and an example of a\ndifferentiable function F colon R\→ R (which can be nowhere monotone) and of\ncompact perfect mathfrakX\⊂ R such that F'(x)=0 for all x\∈\n mathfrakX while F[ mathfrakX]= mathfrakX; thus, the map\n mathfrakf=F restriction mathfrakX is shrinking at every point while,\nparadoxically, not globally. We also present a new short and elementary\nconstruction of em everywhere differentiable nowhere monotone h colon\n R\→ R / and the proofs (not involving Lebesgue measure/integration theory)\nof the theorems of Jarn ' i k and of Laczkovich. The main part of this\nexposition, concerning continuity and first order differentiation, is presented\nin an narrative that answers two classical questions: \To what extend a\ncontinuous function must be differentiable? and \How strong is the\nassumption of differentiability of a continuous function?\n In addition, we overview the results concerning higher order differentiation.\n This includes the Whitney extension theorem and the higher order\ninterpolation theorems related to Ulam-Zahorski problem. Finally, we discuss\nthe results concerning smooth functions that are independent of the standard\naxioms ZFC of set theory. We close with a list of currently open problems\nrelated to this subject.\n