探讨一类函数的性质及其应用
On the Properties of a Class of Functions and Their Applications
范文青1
作者信息
- 1. 闽江师范高等专科学校 福建福州 350108
- 折叠
摘要
首先讨论了初等函数f(x)=xa sinxb在x=0处的连续性和可微性,给出了定理1和定理2及其证明.其次讨论了函数f(x)=xa sinxb在无穷远处的性质,给出定理3函数f(x)在[1+¥)上一致连续的充要条件,并在此基础上给出一个推论和两个性质.最后利用这些性质,探讨函数f(x)=xa sinxb在不同领域的一些应用,给出定理4、定理5和定理6并加以证明.其中,包括函数一致连续性充分条件的讨论、函数在一点处4个Dini导数的任意性,以及函数f(x)在拓扑学中的一个应用.
Abstract
The paper begins with a discussion of the continuity and differentiability of elementary functions f(x)=xasinxbat x=0,giving Theorems 1 and 2 and their proofs.Next,the properties of functions at infinity are discussed,and Theorem 3 is given as a necessary and sufficient condition for their uniform continuity on[1,+∞),based on which a corollary and two properties are given.Finally,these properties are used to explore some applications of the func-tions in different fields,giving the corresponding three theorems and their proofs.These include a discussion of suffi-cient conditions for uniform continuity,the arbitrariness of the four Dini numbers of a function at a point,and an application of function f(x)to topology.
关键词
三角函数/幂函数/初等函数/一致连续Key words
Trigonometric functions/Power functions/Elementary functions/Uniform continuity引用本文复制引用
出版年
2024
NETL
NSTL
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