False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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.
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A pointwise-defined sum of smooth functions need not be smooth

Statement

False claim: whenever $\sum_{n} f_{n} (x)$ is pointwise defined and each $f_{n}$ is smooth, the sum is automatically smooth.

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 definability, is the hypothesis that forces a smooth sum (A locally finite sum of smooth functions is smooth).

Refutation

technique · direct

1.1

For each fixed $x$ , only finitely many of the values $f_{n} (x)$ are nonzero, so $F (x) : = \sum_{n \geq 1} f_{n} (x)$ is pointwise defined; moreover $F (0) = 0$ and $F (1 / n) \geq 1$ for every $n \geq 1$ .

given

2.1

The sequence $1 / n \to 0$ but $F (1 / n) \to̸ F (0)$ , so $F$ is not continuous at $0$ and therefore not smooth.

step 1.1

3.1

This refutes the claim and shows 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)