Symmetric porosity, dimension, and derivates
type: Item
instance of: scholarly article
title: Symmetric porosity, dimension, and derivates
zbMATH Open document ID: 0755.26001
author: Robert W. Vallin
published in: Real Analysis Exchange
publication date: 1992-09-26T00:00:00Z
review text: If \(A\subset R\) \((R\) -- the real line) and \(x\in R\), then the symmetric porosity of \(A\) at \(x\) is defined as \(\limsup_{r\to 0+}\lambda(A,x,r)/r\), where \(\lambda(A,x,r)\) is the supremum of all positive numbers \(h\) such that there is a positive number \(t\) with \(t+h\leq r\) such that both of the intervals \((x-t-h,x-t)\) and \((x+t,x+t+h)\) lie in the complement of \(A\). A set \(A\) is symmetrically porous if it has positive symmetric porosity at each of its points and is called strongly symmetrically porous if it has symmetric porosity one at each of its points. The author constructs a strongly symmetrically porous closed set \(E\subset[0,1)\) which has Hausdorff dimension one. It is shown that there is a continuous monotone function \(f:[0,1]\to R\) such that \(f\) is differentiable at each \(x\in[0,1)-E\) and \(f^ -(x)\neq f^ +(x)\) at each \(x\in E-\{0\}\) \((f^ -(x)\) \([f^ +(x)]\) --- the upper left [right] derivate of \(f\) at \(x)\).
reviewed by: Pavel Kostyrko
Mathematics Subject Classification ID: 26A03
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 54510
zbMATH Keywords: derivates
zbMATH Keywords: positive symmetric porosity
zbMATH Keywords: strongly symmetrically porous closed set
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 54510