Category:Darboux Sums
This category contains results about Darboux Sums.
Definitions specific to this category can be found in Definitions/Darboux Sums.
Upper 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 supremum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$.
Then:
- $\ds \map {U^{\paren f} } P = \sum_{\nu \mathop = 1}^n M_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$
is called the upper Darboux sum of $f$ on $\closedint a b$ belonging (or with respect) to (the subdivision) $P$.
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) $P$.
Subcategories
This category has the following 2 subcategories, out of 2 total.
L
- Lower Darboux Sum (empty)
U
- Upper Darboux Sum (empty)