DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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 injective holomorphic maps

Definition

Let $U \subseteq C$ be open, let $f : U \to C$ be holomorphic (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions), and let $a \in U$ . The map $f$ is locally injective at $a$ if there is a neighbourhood $V$ of $a$ in $U$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open) such that the restriction $f ∣_{V}$ is injective (Injection, surjection, bijection). It is locally injective on $U$ if it is locally injective at every point of $U$ .

The neighbourhood may always be replaced by a smaller open disc centred at $a$ , so this convention agrees with the usual disc formulation.

Depends on

Used by

Holomorphic inverse function theorem and local-degree criterion Theorem

Dependency tree · two levels

10 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

B. V. Shabat, Introduction to Complex Analysis, §1.2 (standard reference, not scraped)