关于函数的一致连续性

On uniform continuity of functions

R的子集D称为具有性质L,如果任一覆盖D的开区间族有Lebesgue数。证明了如果R的子集D具有性质L,则定义在D上的连续函数是一致连续的。作为上述结果的一个推论,改进了通常关于函数一致连续性的一个结果,得到了R的紧致子集(闭区间)上的连续函数是一致连续的。

A subset D of R is called to have property L, if there is a Lebesgue number for every family of open interval covering D. In this paper, we prove that a continuous function defined on a subest D with property L of R is uniformly continuous. This improve a result on uniform continuity of functions, and we obtain that continuous functions defined on compact subsets of R(closed intervals) are uniformly continuous.