CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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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.

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1/sin(1/z) has a nonisolated singularity at 0

Statement refuted

Refuted claim: every singularity at a point of a complex function is automatically isolated.

The witness is

$f (z) = \frac{1}{\sin (1 / z)}$

at $a = 0$ . The function has poles at points tending to $0$ , so the singularity at $0$ is not isolated.

Facts & Assumptions

Given: The function $f (z) = 1 / \sin (1 / z)$ .

[L1]

The zeros of sine are exactly the integral multiples of $π$ (The zero sets of sine and cosine and the least positive common period 2 pi).

Counterexample

technique · direct

1.1

For every positive integer $n$ , the point $z_{n} : = 1 / (n π)$ satisfies $\sin (1 / z_{n}) = \sin (n π) = 0$ by [L1], so $f$ has a pole at each $z_{n}$ .

L1algebra

2.1

The sequence $(z_{n})$ tends to $0$ , so every punctured neighbourhood of $0$ contains some pole of $f$ . Hence no punctured neighbourhood of $0$ is a region on which $f$ is holomorphic, and the singularity at $0$ is not isolated.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

David Greenfield, Rutgers Math 403 diary (standard reference, not scraped)