半连续函数的性质与应用

摘要

函数的种类极为复杂. 在函数论中, 连续函数的性质和应用占有相当重要的地位. 有一类函数虽然不连续, 但却具有一些与连续函数相近的性质, 即连续函数的一个推广——半连续函数. 从而得到了比连续函数更广泛的一类函数的性质.

通过对半连续函数的研究, 对半连续函数在数学分析中的应用奠定了理论基础. 首先简述连续函数的性质与应用, 之后重点讨论半连续函数的性质, 详细介绍运算性, 保号性, 以及拓扑空间上半连续函数性质定理. 推广到紧致空间中半连续函数的应用. 最后辨析连续函数与半连续函数性质、应用, 最终应用连续函数性质解决半连续函数的问题.实际上半连续函数理论在古典分析和现代分析中都有着较为广泛的应用. 比如在最优化问题、变分不等式问题、相补问题及对策论问题都有着举足轻重的作用.

关键词：半连续；连续；函数

Abstract

Category of function is very complicated. Characterization and application of continuous functions are very important in the function theory. Although a kind of function is also continuous, its characterization is similar with the continuous functions, which is called extension of the continuous functions semi-continuous functions, thus a kind of function with more winder characterization is obtained.

Through the study, half of the continuous function in the mathematical analysis continuous function which lay a theoretical foundation for the application. First, this paper expounds the nature of the continuous function and application, and then discusses the nature of the semi-continuous functions, detailed mathematical and application, introduced the number of topological space, and the first half of the continuous function theorem of generalized to nature. Tight space in the application of semi-continuous functions. Finally differentiate continuous function and semi-continuous functions properties, application, and finally application continuous function semi-continuous functions nature solution of the problem. Half a continuous function in the classical theory analysis and modern analysis has a wide range of applications. For example, in the most problems, variational inequalities, phase problems and countermeasures for the theory of and so on all has a pivotal role.

Key words:semi－continuous;continuous;functions;

目录

摘要............................................................................................................................................. I Abstract ...................................................................................................................................... I I

绪论 (1)

第1章连续函数2 1.1 连续函数的性质2

1.1.1连续函数的局部性质及应用2

1.1.2闭区间上连续函数的基本性质3

1.1.3 一致连续性及其应用4第2章半连续函数7

2.1上下半连续函数的性质7

2.1.1运算性质及应用7

2.1.2保号性及应用8

2.1.3无介值性8

2.1.4函数的界8

2.1.5 内闭区间上有界9

2.1.6 保半连续性10 2.2 拓扑空间上半连续函数的性质12

2.2.1 运算性质及其应用13

2.2.2 确界性质及其应用14

2.2.3紧致空间上的半连续函数15

2.2.4 长度的半连续性15第3章半连续函数的异同17 3.1 半连续函数与连续函数的比较17 3.2 半连续函数与连续函数区别18第4章运用连续函数解决半连续函数问题 (20)

结论 (22)

参考文献 (23)

致谢 (24)