Nowhere monotone functions and microscopic sets

被引：5

作者：

Karasinska, A. ^{[1
]}

Wagner-Bojakowska, E. ^{[1
,2
]}

机构：

[1] Univ Lodz, Fac Math & Comp Sci, PL-90238 Lodz, Poland

[2] Coll Comp Sci, Chair Math, PL-93008 Lodz, Poland

来源：

ACTA MATHEMATICA HUNGARICA | 2008年 / 120卷 / 03期

关键词：

nowhere monotone function; one-to-one restriction; microscopic set;

D O I：

10.1007/s10474-008-7093-y

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We investigate how large a set can be on which a continuous nowhere monotone function is one-to-one. We consider the sigma-ideal of microscopic sets, which is situated between the countable sets and the sets of Hausdorff dimension zero and prove that the typical function in C[0, 1] (in the sense of Baire) is nowhere monotone and one-to-one except on some microscopic set. We also give an example of a continuous nowhere monotone function of bounded variation on [0,1], which is one-to-one except on some microscopic set; so it is not a typical function.

引用

页码：235 / 248

页数：14