False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.
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FALSE: Goursat's triangle conclusion requires a separate continuity hypothesis on $f^{'}$

Statement

False claim: To conclude that a holomorphic function has zero integral around the boundary of every filled triangle in its open domain, one must separately assume that its derivative is continuous.

Facts & Assumptions

Given: The asserted need for a separate continuity hypothesis on the derivative.

[L1]

Goursat's triangle theorem assumes only that $f$ is holomorphic on an open set containing the filled triangle and concludes that its boundary integral is zero; it explicitly makes no continuity assumption on $f^{'}$ (Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain).

Refutation

technique · direct

1.1

Under the holomorphy and containment hypotheses, [L1] already gives the claimed zero boundary integral.

2.1

Since continuity of $f^{'}$ is absent from the hypotheses of [L1], it is not a separately required assumption for that conclusion, and the claim is false.

step 1.1L1∎

Depends on

Goursat's triangle theorem: a holomorphic function integrates to zero around every triangle contained in its domain

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Stein and R. Shakarchi, Complex Analysis, Ch. 2, Theorem 1.1 (standard reference, not scraped)
Lars Ahlfors, Complex Analysis, third edition, Ch. 4, Section 1.4 (standard reference, not scraped)

FALSE: Goursat's triangle conclusion requires a separate continuity hypothesis on f′f′