Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable

Depends on

• Riemann's criterion on a nondegenerate rectangle in $\mathbb{R}^m$: integrability is equivalent to arbitrarily small Darboux gaps
• Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$
• Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh
• Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
• Heine-Borel in $\mathbb{R}^n$: with the Euclidean metric a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
• Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
• Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
• Open cover, subcover, compact metric space, and compact subset of a metric space
• The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$
• Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page
• The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
• A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
• Lower bound, bounded below, bounded set

Used by

• A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions Corollary
• A continuous function on a closed rectangle has repeated Riemann integrals in every coordinate order, all equal to its multiple integral Corollary
• The integral of a product function on a product rectangle is the product of the two integrals Corollary
• A deterministic integral construction of a Gaussian process Example
• A normalized compactly supported top form on Euclidean space Example
• A smooth finite-variation integrand and its deterministic error bound Example
• A two-cell linear finite element best approximation for Poisson Example
• Cancellation of the leading Simpson error coefficient Example
• Chebyshev extremal nodes converge to the arcsine equilibrium measure Example
• Convex data bracket the integral between composite midpoint and trapezoid values Example
• Principal value distribution one over x Example
• Simpson rule integrates a cubic exactly on one two-panel block Example
• Simpson rule integrates a quadratic exactly on one two-panel block Example
• The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes Example
• The positive-type Gaussian on the real line and its cyclic model Example
• The Simpson indicator misses a feature between its sample nodes Example
• Borel change of variables from the compact-support formula and Radon uniqueness Lemma
• Convergence of dyadic tensor composite-Simpson rules Lemma
• Finite chart localization gives choice-free integration and compact Stokes Lemma
• Finite-order analytic cutoff sequence Lemma
• Leading composite Simpson error coefficient Lemma
• Midpoint panel error kernel and second-derivative bound Lemma
• Mixed-derivative formula for smooth Hardy–Krause variation Lemma
• Peano kernel identity for one Simpson panel Lemma
• Stokes identity on a rectangular Yee face Lemma
• The compact-support Fourier-Laplace transform is entire in each coordinate Lemma
• Zero-integral compactly supported top forms on Euclidean space have compactly supported primitives Lemma
• A jointly continuous finite-interval parameter integral of holomorphic functions is holomorphic Theorem
• The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction Theorem

Dependency tree · two levels

84 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis, Riemann Integral in Several Variables (standard reference, not scraped)
• J. Lebl, Basic Analysis, The Riemann-Lebesgue Criterion (standard reference, not scraped)