Monotonicity properties of Darboux sums
If \(f:I=[a,b]\to\mathbb R\) let \(\underline S_n, \overline S_n\) denote, respectively, the upper and lower Darboux sums associated with a partition of \(I\) into \(n\) equal intervals. It is classical that \(\underline S_n\leq \underline S_{2n}\leq\overline S_{2n}\leq \overline S_n\) and, as the author shows, if \(f\) is continuous monotone convex or concave the sequences \(\underline S_n, \overline S_n\) are monotone. However as is shown in this paper this is not true in general. In fact the set of functions for which the sequences are monotone is thin in the following sense: let \(X\) denote the Banach space \({\mathcal C}(I)\) given the sup norm and \(U=\{f: f\in X\land\; \underline S_{n+1}\leq \underline S_n\leq\overline S_n\leq \overline S_{n+1} \text{ for an infinity of } n\in \mathbb N\}\) then \(U\) is a dense \({\mathcal G}_{\delta}\) in \(X\). The author further proves that this is the case if we restrict attention to monotone functions in \(X\).
scientific article; zbMATH DE number 5778785