| 梁永顺,张 琦.含有不可数个无界变差点的一维连续函数[J].数学年刊A辑,2018,39(2):145~152 |
| 含有不可数个无界变差点的一维连续函数 |
| 1-Dimensional Continuous Functions with Uncountable Unbounded Variation Points |
| Received:May 27, 2016 Revised:May 17, 2017 |
| DOI：10.16205/j.cnki.cama.2018.0013 |
| 中文关键词: Cantor集, 盒维数, 变差, 图像长度 |
| 英文关键词:Cantor set, Box dimension, Variation, Length of graph |
| 基金项目:本文受到国家自然科学基金(No.11201230, No.11271182)和江苏省自然科学基金(No.BK20161492)的资助. |
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| 中文摘要: |
| 在单位区间$[0,1]$ 上构造了图像长度为无穷的一维连续函数. 该函数含有不可数个但Lebesgue测度为$0$的无界变差点.
所有无界变差点组成的集合中每一点皆为该集合的聚点. |
| 英文摘要: |
| A 1-dimensional continuous function whose graph has infinite length on $[0,1]$ has been constructed.
Unbounded variation points of this function are uncountable, while Lebesgue measure of them is 0. All unbounded variation points
are accumulation points of the set of unbounded variation points of the function. |
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