函数的一致连续性及其应用
关键词：连续；一致连续；函数
Uniform Continuity of Function and Their Application
Abstract： It is an important concept to describe the whole nature of the uniform continuity of the function,it plays an important role for the study of function in mathematics analysis. This article elaborates the essence from the perspective of the geo-metric meaning and conception. This paper presented a series of decision theorems of uniform continuity of derivable function. The comparability methods of function’s uniform continuity were established,which made it possible to distinguish the uniform continuity of complex function through comparing it with the uniform continuous function.This paper will study the fundamental operations on the uniform continuity with binary functions, and analyzing with examples the relationships between uniform continuity and boundness of function, between uniform continuity and boundness of partial derivative, between uniform continuity and boundness of direction derivative , between uniform continuity and total differential for binary functions. Lastly, it is proved that the necessary and sufficient conditions for that an operator is uniform continuity.
Key words: continuous; uniformly continuous；function
目 录
摘 要. 1
引言 2
1. 一元函数一致连续性的相关概念和几何意义 3
1.1一元函数一致连续性的相关概念 3
1.2一元函数一致连续的几何意义 3
2. 一元函数一致连续性的判定 5
2.1一元函数一致连续性的常用判定定理 5
2.2利用导数判定一元函数一致连续性 5
2.3一元函数一致连续性的比较判别法 6
3. 二元函数的一致连续 10
3.1二元函数一致连续性的相关概念 10
3.2二元一致连续函数的四则运算 10
3.3二元函数一致连续性的几个注释 11
3.4函数的一致连续性与方向导数和可微之间的关系 . 13
4. 线性赋范空间上算子的一致连续性 15
4.1算子的一致连续性的相关概念 15
4.2连续算子一致连续的两个充分条件 15
4.2算子一致连续的几个充要条件 16
5.结束语 18
参考文献 19
致谢 20
函数的一致连续性及其应用引言
函数的一致连续性概念是数学分析中的一个重要概念,它反映的是函数的整体性质, 是连续函数理论的重要组成部分,然而它却没有像连续函数和可导函数那样直观和明了的几何意义, 所以对一致连续的理解只能从定义的字面上掌握, 对一致连续的实质则很难深刻理解.所以，对函数一致连续性的判定是数学分析中一个较为困难的问题,一般地,都是按照定义或教材中常用的定理判定．用定义判定比较复杂,而常用的数学分析教材常常使用的判定定理又局限于定理的使用条件,寻找较好的判定方法对判定函数一致连续性非常重要．
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