ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Binary endorelation
Preordering relation
Partial ordering relation
Partially ordered set
Closed interval
Implicit interval partition
Implicit basic real interval partition
Closed real interval tagged partition
Stieltjes sum
Definition D5521
Riemann sum
Formulation 1
Let $[a, b] \subset \mathbb{R}$ be a D544: Closed real interval such that
(i) \begin{equation} a < b \end{equation}
(ii) $P = ((x_0, x_1, \ldots, x_N), (x^*_1, \ldots, x^*_N))$ is a D5520: Closed real interval tagged partition for $[a, b]$
(iii) $f : [a, b] \to \mathbb{R}$ is a D4364: Real function on $[a, b]$
The Riemann sum for $f$ with respect to $P$ is the D993: Real number \begin{equation} \mathcal{R}_P(f) : = \sum_{n = 1}^N f(x^*_n)(x_n - x_{n - 1}) \end{equation}
Riemann integrable real function