高等数学教学中一致连续性的育人作用和判定定理  

The Educational Role and Judgment Theorem of Consistency and Continuity in Higher Mathematics Teaching

作　　者：于晟伟 赵赫 何永明 王洪庆 宋云涛 Yu Chengwei;Zhao He;He Yongming;Wang Hongqing;Song Yuntao(China Fire and Rescue Institute,Beijing 102202,China)

机构地区：[1]中国消防救援学院,北京北京102202

出　　处：《人文与社会科学学刊》2025年第3期27-31,共5页

摘　　要：一致连续性证明是高等数学教学的关键难点,学生常因其抽象与逻辑复杂性而理解不深。本文创新性地提出一种简明证明方法,系统探讨函数导数性质与一致连续性及其非一致连续性的内在联系,并总结为一系列定理。深入剖析一致连续性与非一致连续性定义,明确二者在函数行为上的本质差异。分析函数导数的有界性、单调性等关键性质,揭示其对函数一致连续性的影响机制。在此基础上,总结出一套将导数性质与一致连续性判断直接关联的定理体系,为学生提供清晰、系统的判断工具。为增强学生对理论的理解与应用能力,本文引入丰富实例。通过具体函数的分析与证明,直观展示如何借助导数性质判断一致连续性,使学生能够将抽象理论与实际问题相结合。教学实践表明,这种方法显著提升了学生的学习效果。课堂反馈显示,学生对一致连续性的理解更加深入,证明准确性提高,解决相关问题时的自信心和兴趣也有所提升。本研究为高等数学教学提供新思路,有助于增强学生对抽象数学概念的理解与应用能力,推动高等数学教学质量的提升。The proof of consistency and continuity is a key and difficult point in higher mathematics teaching,and students often do not understand it deeply due to its abstraction and logical complexity.This article innovatively proposes a concise proof method that systematically explores the intrinsic relationship between the derivative properties of functions and their consistency and non consistency continuity,and summarizes it into a series of theorems.Thoroughly analyze the definitions of consistent continuity and inconsistent continuity,and clarify the essential differences between the two in terms of function behavior.Analyze the key properties such as boundedness and monotonicity of function derivatives,and reveal their impact mechanism on the uniform continuity of functions.On this basis,a set of theorem systems is summarized that directly relate derivative properties to consistent continuity judgments,providing students with clear and systematic judgment tools.To enhance students'understanding and application ability of theory,this article introduces rich examples.By analyzing and proving specific functions,demonstrate intuitively how to use derivative properties to determine consistency and continuity,enabling students to combine abstract theory with practical problems.Teaching practice has shown that this method significantly improves students'learning outcomes.Classroom feedback shows that students have a deeper understanding of consistency and continuity,demonstrating improved accuracy and increased confidence and interest in solving related problems.This study provides new ideas for higher mathematics teaching,which helps enhance students'understanding and application ability of abstract mathematical concepts,and promotes the improvement of the quality of higher mathematics teaching.

关 键 词：一致连续性证明 高等数学教学 导数性质 教学实践 

分 类 号：G63[文化科学—教育学]