关于函数一致连续性的研究.pdf

关于函数一致连续性的研究 摘 要 一致连续是分析学中的重要概念,一般的数学分析教材只给出函数一致连续性 的定义和有限闭区间连续函数一致连续性的判别法;本文针对传统方法缺少直观性 的缺点,通过研究函数的性质给出较为直观的判别方法;首先对一致连续的主要结论 进行系统整理,主要有:一致连续的复合函数一致连续;函数满足利普希茨条件必一 致连续;函数图象变化陡峭则非一致连续;函数的导函数在区间上连续有界是一致连 续的;导数较大的函数若一致连续,则较小的也一致连续;函数在开区间上单调拟可 导在该区间内处处存在且有界,函数一致连续;开区间内单调有界函数在区间两侧极 限都存在则一致连续等.其次对应一致连续性判别法给出类似的非一致连续性判别 法;最后对常见函数的一致连续性进行分析,弥补了教材缺少直观的计算性判别法之 不足;对一元函数的一致连续性判别法拓展到二元函数领域进行了初步探讨. 关键词:函数,一致连续,非一致连续,初等函数 I 关于函数一致连续性的研究 Abstract Uniform continuity is an important concept in analytics. The average mathematical textbooks only provide the definition of continuity of function and the discriminance function method of limited closed interval. This thesis provides a more intuitive approach by studying the nature of function to avoid the lack of intuitive judgment in traditional methods. Firstly, this thesis coordinate the main conclusions of uniform continuity in line systematically, which includes uniformly continuous function, the uniformly continuous composite function, the function should satisfy the Lipschitz condition, steep change of function image is for non-uniform, mediated function in the interval bounded is uniform continuity, derivative of the function if the larger uniformly continuous, then the smaller is also uniformly continuous, function in the open interval on the quasi-monotone in the range derivative exists and is everywhere bounded, uniformly continuous function; open range bounded monotone function in the interval, there are limits on both sides of the same continuous, the second criterion corresponds to the same given the continuity of the non-uniform co