Strong porosity features of typical continuous functions
type: Item
instance of: scholarly article
title: Strong porosity features of typical continuous functions
zbMATH Open document ID: 0592.26010
DOI: 10.1007/BF01955016
published in: Acta Mathematica Hungarica
publication date: 1985-01-01T00:00:00Z
review text: A subset A of R is called bilaterally strongly porous at a point a, \(a\in R\), iff the right-porosity \(p_+(A,a)\) and the left-porosity \(p_- (A,a)\) of A at a are both 1. The right-porosity \(p_+(A,a)\) of A at a is the number: \(\limsup_{t\to 0+}\quad d(A,a,t)/t,\) where d(A,a,t) is the length of the largest open interval in \((x,x+t)-A.\) Their main result is the Theorem 2.5: For each \(\sigma\)-compact subset K of the space \(C(<0,1>)\) of all continuous functions on \(<0,1>\) with the supremum norm, the set \(F=\{f\in C(<0,1>):\) for each \(g\in K\) the set \(\{x\in <0,1>:\quad f(x)=g(x)\}\) is bilaterally strongly porous\(\}\) is residual in \(C(<0,1>)\). The authors give also three consequences of their result and concluding remarks. We give here their third consequence: For any system \(E=\{E_ x:\quad x\in <0,1>\}\) of paths, where \(E_ x\) is not bilaterally strongly porous at x for every \(x\in <0,1>\), the class of all functions of \(C(<0,1>)\) which are nowhere E-differentiable is an in \(C(<0,1>)\) residual set.
reviewed by: Ladislav Mišík
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 28A05
zbMATH DE Number: 3950893
zbMATH Keywords: typical continuous function
zbMATH Keywords: bilaterally strongly porous subsets of
zbMATH Keywords: R
zbMATH Keywords: path-differentiation
zbMATH Keywords: nowhere differentiable functions
zbMATH Keywords: residual set
author: Andrew M. Bruckner
author: John Haussermann
MaRDI profile type: Publication
cites work: Current trends in differentiation theory
cites work: The Level Structure of a Residual Set of Continuous Functions
cites work: On the Differentiability Structure of Real Functions
cites work: Path Derivatives: A Unified View of Certain Generalized Derivatives
cites work: Q5733836
cites work: Über die Differenzierbarkeit stetiger Funktionen
cites work: Differentiable restrictions of continuous functions
cites work: Sur les nombres dérivés
cites work: Sur les fonctions non dérivables
cites work: Q3684336
full work available at URL: https://doi.org/10.1007/bf01955016
OpenAlex ID: W2018215551
description: scientific article; zbMATH DE number 3950893