Upper and lower Darboux sums

Definition

Let $a < b$, let $f \colon [a, b] \to \mathbb{R}$ be bounded, and let $P = (x_0, \dots, x_n)$ be a partition of $[a, b]$. For each $i$, let $M_i = \sup f([x_{i-1}, x_i])$ and $m_i = \inf f([x_{i-1}, x_i])$. The upper and lower Darboux sums of $f$ for $P$ are $$U(f, P) = \sum_{i=1}^{n} M_i (x_i - x_{i-1}), \qquad L(f, P) = \sum_{i=1}^{n} m_i (x_i - x_{i-1}).$$