Monotonicity Properties of Darboux Sums

Abstract

Let $f:[a, b]\to \R$ be a continuous function. Dividing the interval $[a, b]$ into subintervals of equal length, we obtain partitions of $[a, b]$ for which the upper and lower Darboux sums of $f$ constitute two sequences, which converge to the definite integral of $f$ in $[a, b]$ from above and below respectively. We study the monotonicity properties of these sequence and we prove that their non-monotonicity is a generic (quasi-sure) property in the space $C([a, b])$.