函数连续性概念的历史:从欧拉到勒贝格  

The Historical Development of the Concept of Function's Continuity:From Euler to Lebesgue

作　　者：李亚亚 王昌[2] LI Yaya;WANG Chang(School of Mathematics,Xi’an University of Finance and Economics,Xi’an,Shaanxi,710100;Institute for Advanced Studies in History of Science,Northwest University,Xi’an,Shaanxi,710127)

机构地区：[1]西安财经大学数学学院,陕西西安710100 [2]西北大学科学史高等研究院,陕西西安710127

出　　处：《自然辩证法通讯》2024年第2期61-67,共7页Journal of Dialectics of Nature

基　　金：国家自然科学基金项目“算子理论的若干历史问题研究”(项目编号:12301002);陕西省自然科学基金项目“希尔伯特空间理论的历史研究”(项目编号:2021JQ-763);陕西省教育厅专项基金项目“紧算子理论的历史研究”(项目编号:19JK0317)。

摘　　要：函数是数学中的一个基本概念,连续性是函数的重要性质之一。函数连续性概念源于几何直观,受函数为解析表达式概念的限制,函数的连续性一开始被看成是解析法则的不变性。随着函数概念的发展,并顺应分析学严格化发展的潮流,柯西将连续意味着不间断的几何观念用精确的分析学定义表述出来,从而给出了现代意义下的函数连续性,最终魏尔斯特拉斯给出了现行的ε-δ定义。对函数连续性概念演变过程的研究可以为反观分析学严格化历程提供一个窗口。The function is a basic concept in mathematics,and continuity is one of its important properties.The concept of a function's continuity stems from geometric intuition.Because a function was defined as analytical expressions,a continuous function was initially seen as one determined by a single analytical law.Following the trend towards rigor in analysis,Cauchy gave the concept of a function's continuity in the modern sense by formulating the geometric notion of continuity.Based on Cauchy's work,Weierstrass gave the current definition of a function's continuity.The historical exploration on the evolution of the concept of a function's continuity can provide a window into the rigorous process of analysis.

关 键 词：分析学严格化 函数 连续性 可微性 可积性 

分 类 号：N09[自然科学总论—科学技术哲学] O1[理学—数学]