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
Riemann sum
Definition D1760
Riemann integrable real function
Formulation 1
Let $[a, b] \subset \mathbb{R}$ be a D544: Closed real interval such that
(i) \begin{equation} a < b \end{equation}
(ii) $f : [a, b] \to \mathbb{R}$ is a D4364: Real function on $[a, b]$
Then $f$ is a Riemann integrable real function on $[a, b]$ if and only if \begin{equation} \exists \, R \in \mathbb{R} : \forall \, \varepsilon > 0 : \exists \, \delta > 0 \left( P \text{ is a tagged partition for } [a, b] \text{ with } \text{Mesh}(P) < \delta \quad \implies \quad \left| \mathcal{R}_P(f) - R \right| < \varepsilon \right) \end{equation}