On the box dimensions of graphs of typical continuous functions

被引：29

作者：

Hyde, J. ^{[1
]}

Laschos, V. ^{[2
]}

Olsen, L. ^{[1
]}

Petrykiewicz, I. ^{[3
]}

Shaw, A. ^{[1
]}

机构：

[1] Univ St Andrews, Dept Math, St Andrews KY16 9SS, Fife, Scotland

[2] Univ Bath, Dept Math Sci, Bath BA2 7AY, Avon, England

[3] Univ Grenoble 1, Inst Fourier, F-38402 St Martin Dheres, France

来源：

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS | 2012年 / 391卷 / 02期

关键词：

Box dimension; Continuous function; Baire category;

D O I：

10.1016/j.jmaa.2012.02.044

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Let X subset of R be a bounded set; we emphasize that we are not assuming that X is compact or Borel. We prove that for a typical (in the sense of Baire) uniformly continuous function f on X, the lower box dimension of the graph of f is as small as possible and the upper box dimension of the graph of f is as big as possible. We also prove a local version of this result. Namely, we prove that for a typical uniformly continuous function f on X, the lower local box dimension of the graph of f at all points x is an element of X is as small as possible and the upper local box dimension of the graph of f at all points x is an element of X is as big as possible. (C) 2012 Elsevier Inc. All rights reserved.

引用

页码：567 / 581

页数：15