On the Properties of a Class of Functions and Their Applications
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.
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