False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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).

[L1]

Euclidean Whitney approximation is formulated with such pointwise positive error functions (Whitney approximation for Euclidean-valued maps).

Refutation

technique · direct

1.1

The function $ε (x) = e^{- ∣ x ∣}$ tends to $0$ as $∣ x ∣ \to \infty$ , so the requirement $∣ G (x) - F (x) ∣ < ε (x)$ demands finer and finer control at infinity. No single constant $η > 0$ can encode that condition, because for large $∣ x ∣$ one has $ε (x) < η$ .

F1givenalgebra

2.1

The correct global theorem [L1] is therefore phrased with variable positive error functions rather than one uniform tolerance. That is exactly what allows the approximation scale to shrink along different ends of a noncompact source.

L1step 1.1

3.1

Hence the claim that one global uniform bound is always the right notion is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)