Definition:Upper Sum

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Definition

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a bounded real function.


Let $P = \left\{{x_0, x_1, x_2, \ldots, x_n}\right\}$ be a finite subdivision of $\left[{a \,.\,.\, b}\right]$.

For all $\nu \in \left\{{1, 2, \ldots, n}\right\}$, let $M_\nu^{\left({f}\right)}$ be the supremum of $f$ on the interval $\left[{x_{\nu - 1} \,.\,.\, x_{\nu}}\right]$.


Then:

$\displaystyle U^{\left({f}\right)} \left({P}\right) = \sum_{\nu \mathop = 1}^n M_\nu^{\left({f}\right)} \left({x_{\nu} - x_{\nu - 1}}\right)$

is called the upper sum of $f$ on $\left[{a \,.\,.\, b}\right]$ belonging (or with respect) to (the subdivision) $P$.


If there is no ambiguity as to what function is under discussion, $M_\nu$ and $U \left({P}\right)$ are often seen.


Also known as

The notation $U \left({P, f}\right)$ can be used in place of $U^{\left({f}\right)} \left({P}\right)$.


The upper sum is also known as the upper Darboux sum or the upper Riemann sum.


Also see


Sources