基于实值函数极限理论的无穷大符号讨论  

The sign of infinity based on the theory of limit of real-valued functions

作　　者：支元洪[1] 何青海 ZHI Yuan-hong;HE Qing-hai(School of Mathematics and Statistics,Yunnan University,Kunming 650500,China)

机构地区：[1]云南大学数学与统计学院

出　　处：《云南大学学报（自然科学版）》2019年第6期1090-1100,共11页Journal of Yunnan University(Natural Sciences Edition)

基　　金：国家自然科学基金(11261067)

摘　　要：比较了实数系的仿射扩充和射影扩充2种扩充方式,研究并总结了在实分析,特别是实值函数极限论中引入有符号无穷大-∞,+∞以及无符号无穷大∞的好处与弊端,从实数扩充角度得到如下结论:在实值函数极限理论中可不引入x→x0limf(x)=∞,从而避免记号的混淆,且使得只使用+∞和-∞的实值函数极限理论更精细化,更能方便且准确地刻画函数在极限点附近的局部分析性质,也使整个实值函数极限理论更清晰和统一,进而使基于极限的实分析的一些重要定理之推广形式的证明更简洁,应用范围更广.By comparing two methods of extension of real numbers,that is,the affine extension,and the projective extension,together with studying and summarizing the advantages and drawbacks when introducing,in real analysis,especially in the limit theory of real-value functions,the signed infinities -∞,+∞ and the unsigned infinity ∞.It is obtained that,without introduction of x→x0limf(x)=∞,there are,in real analysis,no confusions of notation for limit expression any longer,and the limit theory of real-valued functions is more precise,by just using only +∞ and -∞,and hence,the depiction of local properties of functions near the limit point is easier and more precise.Moreover,without unsigned infinity,the whole limit theory of real-valued functions is more clear and unified,which make it possible to generalize some important theorems in real analysis,with neat proofs and multitude of applications.

关 键 词：仿射扩充实数系 射影扩充实数系 有符号无穷大 无符号无穷大 局部保序性 夹挤原理 紧性 

分 类 号：O171[理学—数学] O174.11[理学—基础数学]