闭区间上连续函数的性质及其推广摘要：在我们的学习生活中，常常会遇到最最基本的一种函数-连续函数，在初高中的数学中，我们遇到的函数几乎都是连续函数，这类基本的函数却有很多优良的性质，尤其是闭区间上的连续函数，它更是具有很多开区间所不具备的性质。故了解闭区间上的连续函数是很有必要的，为了解闭区间上连续函数的性质，我们得先从什么是连续函数说起，要想知道什么是连续函数，就得先了解函数极限，故本文从函数极限说起，介绍了函数极限的定义，函数极限的性质，函数的连续性定义，连续函数的性质，再介绍闭区间上连续函数的性质，最后列举了有关闭区间上连续函数性质的推广，如最值定理的推广，介值定理的推广，有界定理的推广，零点定理的推广，根的存在定理的推广等，并对以上这些定理的推广进行分析、比较，本文将从两方面进行比较，一是从条件比较，二是从结论比较，最后总结出闭区间上连续函数性质的推广的一些关键词：闭区间上连续函数性质的推广闭区间上连续函数的性质连续函数The properties and generalization of continuous function on closed intervalAbstract In our study and life,we often encounter the most basic function-continuousfunction.In the mathematics of junior and senior high school,almost all of the functions weencounter are continuous functions,but these basic functions have many excellent properties,especially the continuous function on the closed interval,which has many properties that theopen interval does not have.Therefore,it is necessary to understand the continuous functionon the closed interval.In order to solve the properties of the continuous function on the closedinterval,we must start from what is the continuous function,and to know what is thecontinuous function,we must first understand the function limit.So this paper starts from thefunction limit,introduces the definition of the function limit,the properties of the function limit,the definition of the continuity of the function,and the properties of the continuous functionQuality,then introduce the properties of continuous function on closed interval,and finallyenumerate the generalization of continuous function properties on closed interval,such as thegeneralization of the most value theorem,the generalization of intermediate value theorem,the generalization of bounded theorem,the generalization of zero point theorem,thegeneralization of the existence theorem of root,etc.,and analyze and compare thegeneralization of these theorems.This paper will compare them from two aspects The secondis the conclusion comparison.Finally,some conclusions about the extension of continuousfunction properties on closed interval are summarized.Key words:The extension of the properties of continuous function on closed intervalproperties of continuous functions on closed intervals continuous function目录引言1.预备知识…1.1.1函数在一点的极限.1.1.2函数极限的性质.11.2函数的连续性41.2.1连续性概念1.2.2左连续和右连续1.2.3间断点及其分类.1.3区间上的连续函数…1.3.1狄利克雷函数的连续性1.3.2黎曼函数的连续性…71.4连续函数的性质81.4.1连续函数的局部性质82.闭区间上连续函数的基本性质2.1有界性定理2.2最值定理.2.3零点存在定理…2.4介值性定理…2.5根的存在定理.133.闭区间上的连续函数性质的推广及其比较3.1介值性定理的推广及其比较….133.1.1介值性定理的推广133.1.2介值性定理推广的比较…3.2有界性定理的推广及其比较…183.2.1有界性定理的推广…183.2.2有界性定理推广的比较.193.3最值定理的推广及其比较..19