CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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 locally constant step map on the disconnected open set $\mathbb{R}\setminus\{0\}$ has zero total derivative but is not globally Lipschitz

Statement refuted

A uniform total-derivative bound on every open domain implies a global Lipschitz bound on that domain.

Facts & Assumptions

Given: $U=\mathbb R\setminus\{0\}$ and $f:U\to\mathbb R$ defined by $f(x)=0$ for $x<0$ and $f(x)=1$ for $x>0$ .

[L1]

In the total-derivative definition, the normalized remainder tends to zero as $h$ tends to zero (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\|h\|_2)$ remainder).

[L2]

A map is Lipschitz with constant $L$ when $|f(x)-f(y)|\le L|x-y|$ for every pair of points in its domain (Lipschitz map, $\alpha$ -Hölder map for rational $0 < \alpha \le 1$ , and contraction).

Counterexample

technique · direct

1.1

Each $x\in U$ has a small interval contained in its own component of $U$ , on which $f$ is constant; hence $Df(x)=0$ by [L1].

L1L2

2.1

For any $L\ge0$ , take $t=1/(2(L+1))$ . The points $-t,t\in U$ satisfy $|f(t)-f(-t)|=1>2Lt=L|t-(-t)|$ , so [L2] fails for that $L$ .

step 1.1L2algebra

3.1

The segment from $-t$ to $t$ contains $0\notin U$ , so $U$ is not convex; this is exactly the omitted hypothesis of the mean-value inequality.

step 1.1step 2.1∎

Depends on

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: 125 results over 23 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

J. Lebl, Basic Analysis I, §8.4 (standard reference, not scraped)

A locally constant step map on the disconnected open set R∖{0}\mathbb{R}\setminus\{0\}R∖{0} has zero total derivative but is not globally Lipschitz