| 肖飞,唐小军,黄紫橙.原函数存在性的相关研究及其应用实例[J].井冈山大学自然版,2025,46(3):15-20 |
| 原函数存在性的相关研究及其应用实例 |
| RESEARCH ON THE EXISTENCE OF ANTIDERIVATIVES AND ITS APPLICATION EXAMPLES |
| 投稿时间：2025-03-01 修订日期：2025-03-26 |
| DOI：10.3969/j.issn.1674-8085.2025.03.003 |
| 中文关键词: 间断点 原函数 导数极限定理 函数可积性 |
| 英文关键词: discontinuity point antiderivative derivative limit theorem integrability of functions |
| 基金项目:国家自然科学基金项目(12361036)；井冈山大学博士科研启动项目(JZB1921)；井冈山大学教学改革研究项目(XJJG-20-39，XJJG-23-34) |
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| 中文摘要: |
| 针对具有间断点的函数，研究了该类函数是否存在原函数，并进一步探索了原函数存在性与函数可积性之间的关系。首先，根据高等数学中的间断点相关定义，借助导数极限定理与达布中值定理作为分析工具，对不同类型的间断点函数进行研究，得出其原函数存在性的相关结论：具有第一类间断点的函数，在包含该间断点在内的区间上不存在原函数；具有无穷间断点的函数，在包含该间断点在内的区间上不存在原函数；具有震荡间断点的函数，在包含该间断点在内的区间上可能存在原函数。其次，从函数的不同类型间断点出发，借助数学分析上的两个引理，深入探讨了原函数存在性与函数可积性的关系，并得出两者不是简单的相互推出关系。最后给出了具体的应用实例。 |
| 英文摘要: |
| For functions with discontinuity points, this paper studies whether there exists an original function for this type of function, and further explores the relationship between the existence of the original function and the integrability of the function. Firstly, based on the definition of discontinuity points in advanced mathematics, using the derivative limit theorem and the Darboux's mean value theorem as analytical tools, different types of discontinuity functions are studied, and relevant conclusions about the existence of their original functions are drawn: functions with the first type of discontinuity point do not have the original function on the interval containing the discontinuity point; functions with infinite breakpoints do not have the original function on the interval that includes that breakpoint; functions with oscillatory breakpoints may have the antiderivative function on the interval that includes that breakpoint. Secondly, starting from the different types of discontinuities in functions and using two lemmas in mathematical analysis, the relationship between the existence of the antiderivatives and its integrability is deeply explored，and it is concluded that the two are not simply mutually derivable. Finally, specific application examples are given. |
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