False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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FALSE: continuity of the derivative implies constant rank

Statement

If a Euclidean map has continuous derivative, then its derivative has locally constant rank.

Facts & Assumptions

Given: The polynomial map $f (x, y) = (x, x y)$ .

[L1]

Its derivative has continuous polynomial entries, rank $1$ on $x = 0$ , and rank $2$ on $x \neq 0$ ; every neighbourhood of the origin meets both loci (The map $(x, y) \mapsto (x, x y)$ has nonconstant rank on every neighbourhood of the origin).

[L2]

The correct general conclusion is lower semicontinuity: every rank-at-least- $r$ locus is open (Differential rank is lower semicontinuous).

Refutation

technique · direct

1.1

By [L1], $D f$ is continuous but has two different ranks in every neighbourhood of the origin.

givenL1

2.1

Hence continuity of the derivative does not imply locally constant rank.

step 1.1

3.1

This does not contradict [L2]: the rank- $2$ locus ${x \neq 0}$ is open, so rank jumps upward away from the vertical axis exactly as lower semicontinuity permits.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

J. M. Lee, Introduction to Smooth Manifolds, rank theorem discussion (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Section 11.1 (standard reference, not scraped)