探讨一类函数的性质及其应用

On the Properties of a Class of Functions and Their Applications

首先讨论了初等函数f(x)=x^(a)sinx^(b)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)=x^(a)sinx^(b)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)=x^(a)sinx^(b)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。

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 functions in different fields,giving the corresponding three theorems and their proofs.These include a discussion of sufficient 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.