The Fundamental Theorem of Integral Calculus: a Volterra's generalization applied to flat functions

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).