Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6418413 | Journal of Mathematical Analysis and Applications | 2014 | 13 Pages |
Abstract
This paper examines the level sets of the continuous but nowhere differentiable functionsfr(x)=ân=0ârânÏ(rnx), where Ï(x) is the distance from x to the nearest integer, and r is an integer with râ¥2. It is shown, by using properties of a symmetric correlated random walk, that almost all level sets of fr are finite (with respect to Lebesgue measure on the range of f), but that for an abscissa x chosen at random from [0,1), the level set at level y=fr(x) is uncountable almost surely. As a result, the occupation measure of fr is singular.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Pieter C. Allaart,