TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Locally dominated parameter-dependent improper multiple integrals are continuous

Statement

A locally dominated parameter-dependent improper multiple integral is continuous in the parameter.

More precisely, let $f : D \times I \to R$ be continuous and suppose it is locally dominated near each parameter in the sense of Parameter-dependent improper multiple integrals. Then $F (t) = \int_{D} f (x, t) d x$ is continuous on $I$ in the relative topology.

Facts & Assumptions

Given: The continuous integrand $f$ , parameter interval $I$ , and local domination in the Statement; fix $t_{0} \in I$ .

[L1]

If locally integrable slices $f_{t}$ on an open $D$ satisfy $∣ f_{t} ∣ \leq g$ on a compact parameter set, where $g \geq 0$ and $\int_{D} g < + \infty$ , then for every $η > 0$ one compact Jordan $K \subseteq D$ satisfies $∣ \int_{D} f_{t} - \int_{K} f_{t} ∣ < η$ for every such $t$ (An integrable dominator gives uniform tail control on every compact parameter set).

[L2]

A continuous map from a compact metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[L3]

Proper multidimensional Riemann integrals are monotone and satisfy the absolute-value estimate (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ).

[L4]

The integral of the indicator of a Jordan set equals its content (A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content).

[L5]

An integrable nonnegative dominator makes every locally integrable dominated slice absolutely improperly integrable (Comparison and absolute comparison tests for improper multiple integrals).

[L6]

Every continuous real function on a compact Jordan set is Riemann integrable there (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set).

Proof

technique · direct

1.1

Choose a compact relative parameter neighborhood $C \subseteq I$ of $t_{0}$ and an integrable $g$ dominating $∣ f_{t} ∣$ there. Continuity gives local integrability and [L5] gives absolute improper integrability of the slices. Given $ε > 0$ , [L1] supplies a compact Jordan core $K \subseteq D$ on which both tail errors are below $ε / 3$ .

L1L5choose

1.2

The restriction of $f$ to the compact set $K \times C$ is uniformly continuous by [L2], while [L6] supplies all proper core integrals. Hence, for $t \in C$ sufficiently close to $t_{0}$ , $∣ f (x, t) - f (x, t_{0}) ∣ < ε / (3 (1 + cont K))$ for every $x \in K$ , and [L3] with [L4] makes the compact-core integral difference smaller than $ε / 3$ .

L2L3L4L6

2.1

Split $F (t) - F (t_{0})$ into the two tail errors and the core-integral difference. Steps 1.1 and 1.2 make its absolute value smaller than $ε$ , proving relative continuity at $t_{0}$ , including a one-sided parameter endpoint.

step 1.1step 1.2algebra∎

Depends on

Used by

Differentiation under an improper multiple integral under an integrable derivative bound Theorem

Dependency tree · two levels

41 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

W. F. Trench, Functions Defined by Improper Integrals, Theorem 10 (standard reference, not scraped)