DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.

Positive continuous error functions for strong approximation

Definition

Let $M$ be a smooth manifold. A positive continuous error function on $M$ is a continuous map $ε : M \to (0, \infty) .$ In Whitney approximation on a noncompact manifold, the inequality $∥ G (p) - F (p) ∥ < ε (p)$ is the pointwise fine-control condition that replaces one global uniform error bound.

Depends on

Smooth manifolds and their smooth charts

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., The Whitney Approximation Theorems