积分学的基本定理：Volterra推广到平坦函数
The Fundamental Theorem of Integral Calculus: a Volterra's generalization applied to flat functions
摘要：
在最近的一篇文章[5]中，考虑了一个光滑函数f：[0; 1] --> R的导数在0处为零的情形，并给出了一个整体条件，证明了f确实恒为0。本文的目的是用一个较弱的形式来代替经典的黎曼积分的微积分基本定理，这个定理在文[5]中已经使用过了，它可以追溯到Volterra [7]，但鲜为人知。因此，我们在本文中提出的证明从教学的角度来看也是重要的，只要在文献中有很少的例子，其中明确的下积分和上积分的功能出现（通常的假设，该功能是黎曼可积的要求）。
英文摘要：
In a recent paper [5] a smooth function f : [0; 1] --> R with all derivatives vanishing at 0 has been considered and a global condition, showing that f is indeed identically 0, has been presented. The purpose of this note is to replace the classical Fundamental Theorem of Calculus for the Riemann integral, as it has been used in [5], with a weaker form going back to Volterra [7], which is little known. Therefore the proof we propose in this paper turns to be important also from the teaching point of view, as long as in literature there are very few examples in which explicitly the lower integral and the upper integral of a function appear (usually the assumption that the function is Riemann-integrable is required).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
