CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

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.

Every derivative operator is continuous for locally uniform convergence on holomorphic functions

Statement

For every natural number $k$ , including $k = 0$ , the operator $D^{k} : H (Ω) \to H (Ω), D^{k} (f) = f^{(k)},$ is continuous for local uniform convergence on a plane domain $Ω$ .

Facts & Assumptions

Given: A sequence $f_{n} \to f$ locally uniformly in $H (Ω)$ .

[L1]

A locally uniform limit of holomorphic functions is holomorphic, and every derivative order converges locally uniformly as well (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).

Proof

technique · direct

1.1

Fact [L1] gives $f_{n}^{(k)} \to f^{(k)}$ locally uniformly on $Ω$ for every natural $k$ .

L1given

2.1

This is exactly continuity of the operator $D^{k}$ , and the case $k = 0$ is included because $D^{0}$ is the identity.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Matthias Weber, Complex Analysis, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)