False statementConstruction: AI-adaptedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

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.

FALSE: uniform approximation is the right global notion on every noncompact manifold

Statement

False claim: on every noncompact manifold, one global uniform error bound is the right notion of smooth approximation.

Facts & Assumptions

Given: The continuous function $F (x) = ∣ x ∣$ on $R$ and the positive continuous error function $ε (x) = e^{- ∣ x ∣}$ .

[F1]

A positive continuous error function may vary from point to point (Positive continuous error functions for strong approximation).

Refutation

technique · direct

1.1givenalgebra

Fix any uniform tolerance $η > 0$ , choose $0 < a < η$ , and set $G_{a} (x) = \sqrt{x^{2} + a^{2}}$ . This function is smooth on $R$ and $0 < G_{a} (x) - ∣ x ∣ = \frac{a^{2}}{\sqrt{x^{2} + a^{2}} + ∣ x ∣} \leq a < η$ for every $x$ . Thus it uniformly approximates $F$ .

2.1F1step 1.1algebra

For $x > 0$ , $G_{a} (x) - F (x) = \frac{a^{2}}{\sqrt{x^{2} + a^{2}} + x} \geq \frac{a^{2}}{2 x + a} .$ The last expression eventually exceeds $e^{- x} = ε (x)$ , since $(2 x + a) e^{- x} \to 0$ . Therefore $G_{a}$ fails the pointwise error requirement despite its uniform error being less than $η$ .

3.1step 1.1step 2.1∎

Since this happens for every $η > 0$ on the noncompact manifold $R$ , no fixed uniform tolerance captures the stated fine approximation condition. This refutes the claim.

Depends on

Positive continuous error functions for strong approximation

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., The Whitney Approximation Theorems (standard reference, not scraped)