浅谈函数的一致连续性.doc

浅谈函数的一致连续性 （渤海大学数理学院 辽宁 锦州 121000 中国） 摘要：在数学分析中一致连续函数具有很重要的地位，其定义在数学分析中也算是一个难点。本文主要从一致连续函数的直观理解深入到纯分析的论证，只从一致连续函数本身的性质入手。首先，本文用大量篇幅给出了函数一致连续性的证明并做作比较系统的归纳，把函数一致连续性的证明方法归纳为四个部分：运用区间套定理，致密性定理，覆盖定理以及归结原则四种方法证明了一致连续性定理。其次，本文比较完整的给出了一致连续性函数的判定方法及性质，为我们对一致连续性函数的应用打下了坚实的基础。再次，本文系统、详尽地叙述了一致连续性函数与连续函数的关系，解决了连续函数与一致连续相互转化的问题。最后，介绍了一致连续性函数的描述及其延拓问题。使人们能够对它们有个全面的了解。 关键词：一致连续，一致连续性定理，一致连续性性质，连续函数，一致连续性判定。 Abstract: In the mathematical analysis of uniformly continuous function is a very important position, its definition in the mathematical analysis is also a difficulty. This article mainly from the consistent continuous function intuitive understanding of deep into the pure analysis argument, only from the start with the nature of uniformly continuous function itself. First of all, this paper devotes a lot of space gives the proof of uniform continuity of a function and artificial system are summarized, the proof of uniform continuity of a function methods into four parts: the use of nested interval theorem, compact theorem, covering theorem as well as this principle four methods proved uniform continuity theorem. Secondly, this paper gives a uniformly continuous function determination methods and properties, for us to the uniformly continuity of function application to lay a solid foundation. Again, in this paper, a detailed description of the system of uniform continuity of a function and relation of continuous function, solve the continuous function and the uniform continuity of mutual transformation problem. Finally, introduced the uniform continuity of a function is described and its extension. To enable people to have a comprehensive understanding of their. Key words: Uniform continuity, uniform continuity theorem, uniform continuity properties, continuous function, uniform continuity judgment. 引言 数学分析立足于研究有限维空间的函数分析，它研究了各式各样的函数，其中最重要的一类函数叫做一致连续性函数，它是数学分析乃至整个数学领域的重要部分。一致连续性函数有着许多性质，而实数的连续性和闭区间的紧致性使得闭区间上的连续函数更有着丰富的性质，以下就简单介绍一下一致连续性的定义、定理、判定、性质以及其延拓。 一、一致连续的概念 定义：设函数