CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

A pointwise-finite smooth family whose sum is not continuous

Statement refuted

A pointwise-finite family of smooth functions always has a continuous pointwise sum.

Facts & Assumptions

Given: A smooth bump $η : R \to [0, 1]$ supported in $[- 1, 1]$ with $η (0) = 1$ , and $f_{n} (x) : = η (n^{2} (x - 1 / n))$ for $n \geq 1$ .

[L1]

Local finiteness, not mere pointwise finiteness, is the hypothesis that forces a smooth sum (A locally finite sum of smooth functions is smooth).

Counterexample

technique · direct

1.1

For every fixed $x$ , only finitely many $f_{n} (x)$ are nonzero, so the family $(f_{n})_{n \geq 1}$ is pointwise finite; however $f_{n} (0) = 0$ and $f_{n} (1 / n) = 1$ for every $n \geq 1$ .

given

2.1

The sum $F (x) : = \sum_{n} f_{n} (x)$ therefore satisfies $F (0) = 0$ and $F (1 / n) \geq 1$ for every $n$ , so $F$ is not continuous at $0$ .

step 1.1

3.1

This refutes the statement and exhibits why [L1] needs local finiteness.

L1step 2.1∎

Depends on

A locally finite sum of smooth functions is smooth

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)