CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Zero derivative need not give constancy on a disconnected open set

Statement refuted

If a differentiable real function has derivative zero at every point of an open domain, then it is constant on that domain.

The counterexample below establishes the true witness clause: The total derivative of $f$ is zero at every point of $U$ , but $f$ is not constant on $U$ .

Facts & Assumptions

Given: Let $U = (- 2, - 1) \cup (1, 2)$ and define $f : U \to R$ to equal $1$ on $(- 2, - 1)$ and $- 1$ on $(1, 2)$ .

[L1]

The total derivative $D f (a)$ is the linear map whose remainder in the first-order approximation is little-oh of the displacement (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

Counterexample

technique · direct

1.1

Every point of $U$ has a neighbourhood contained in exactly one component, and $f$ is constant there. The zero linear map therefore leaves a zero remainder in [L1], so $D f (a) = 0$ for every $a \in U$ .

L1given

2.1

The values at $- 3 / 2$ and $3 / 2$ are respectively $1$ and $- 1$ . Thus the total derivative of $f$ is zero at every point of $U$ , but $f$ is not constant on $U$ .

step 1.1given∎

Depends on

The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\|h\|_2)$ remainder

Used by

FALSE: zero derivative on an open set forces constancy False statement

Dependency tree · two levels

5 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

S. Cañez, Northwestern Math 320-2 lecture notes (standard reference, not scraped)