A pathological approximately smooth function
type: Item
instance of: scholarly article
title: A pathological approximately smooth function
zbMATH Open document ID: 0589.26003
DOI: 10.1007/BF01955731
author: Michael J. Evans
author: Paul D. Humke
published in: Acta Mathematica Hungarica
publication date: 1985-01-01T00:00:00Z
review text: A real-valued measurable function f defined on the real line is said to be smooth if at each point x \[ \lim_{h\to 0}(f(x+h)+f(x-h)-2f(x))/h=0. \] It is known that the set of discontinuities of a smooth function is countable; indeed, this set can be characterized as a scattered set. If one replaces the limit in the above definition by an approximate limit, the notion of approximate smoothness results. Although the set of points of approximate discontinuity of an approximately smooth function must be both of measure zero and of first category, an example is constructed here for which the set of points of approximate discontinuity is uncountable.
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 3946499
zbMATH Keywords: approximate discontinuity
zbMATH Keywords: approximately smooth function
MaRDI profile type: Publication
cites work: The continuity of symmetric and smooth functions
cites work: Q3709004
cites work: Symmetric, continuous, and smooth functions
cites work: Smoothness and differentiability in $L_{p}$
full work available at URL: https://doi.org/10.1007/bf01955731
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description: scientific article; zbMATH DE number 3946499