CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-09-09 (Codex)

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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.

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Extension by zero without support away from the boundary is not smooth

Statement refuted

Any smooth function on an open set extends smoothly to the ambient manifold by setting it equal to zero outside the open set.

Facts & Assumptions

Given: The open set $(0, \infty) \subseteq R$ and the smooth function $f (x) = 1$ on it.

[L1]

Smooth maps are continuous (Smooth maps are continuous).

[L2]

Continuity at zero requires the epsilon-delta condition (Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point). Constant functions are smooth: all positive-order coordinate derivatives vanish ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

Counterexample

technique · direct

1.1givenL2

The constant function $f = 1$ is smooth on the open set $(0, \infty)$ by [L2]. Its prescribed zero extension is $F (x) : = 1$ for $x > 0$ and $F (x) : = 0$ for $x \leq 0$ .

2.1L1L2step 1.1

For $ε = 1 / 2$ and every $δ > 0$ , put $x = δ / 2$ . Then $∣ x ∣ < δ$ but $∣ F (x) - F (0) ∣ = 1$ . Thus $F$ is not continuous at zero by [L2], and cannot be smooth by [L1].

3.1step 1.1step 2.1∎

This refutes the universal zero-extension claim. It does not assert that every smooth zero extension requires a cutoff or support separated from the boundary; the counterexample proves only that the recipe fails in general. No choice principle or existence-of-cutoff theorem is used.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)