Category:Definitions/Lower Darboux Sum

This category contains definitions related to Lower Darboux Sum.
Related results can be found in Category:Lower Darboux Sum.

Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \R$ be a bounded real function.

Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$.

For all $\nu \in \set {1, 2, \ldots, n}$, let $m_\nu^{\paren f}$ be the infimum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$.

Then:

$\ds \map {L^{\paren f} } P = \sum_{\nu \mathop = 1}^n m_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$

is called the lower Darboux sum of $f$ on $\closedint a b$ belonging (or with respect) to (the subdivision) $P$.