TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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The Jacobian sign of a regular $C^{1}$ map is constant on a connected domain

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be nonempty, open, and connected, and let $f : U \to R^{n}$ be $C^{1}$ with invertible derivative everywhere. Then either $\det D f (x) > 0$ for every $x \in U$ or $\det D f (x) < 0$ for every $x \in U$ . Thus $f$ has one local orientation throughout $U$ (Local orientation of a regular $C^{1}$ Euclidean map).

Facts & Assumptions

Given: The hypotheses in the Statement. A $C^{1}$ map has continuous derivative-matrix entries (Continuously differentiable maps, local inverses, and local diffeomorphisms), and connected subsets of the real line are order-convex (The connected subspaces of $R$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $R$ ").

[L1]

The function $\det : M_{n} (R) \to R$ is evaluation of a polynomial in the matrix entries (For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries).

[L2]

The continuous image of a connected subset is connected (A continuous image of a connected space is connected, and connectedness is a topological property).

Proof

technique · direct

1.1

The entries of $D f$ are continuous, and [L1] expresses $x \mapsto \det D f (x)$ as a polynomial in them. Hence the Jacobian determinant is continuous on $U$ .

L1given

2.1

By [L2], its image is a connected subset of $R$ . Regularity excludes zero. If the image contained both a negative and a positive value, order-convexity would force it to contain zero, a contradiction. Nonemptiness therefore leaves exactly one sign throughout $U$ .

step 1.1L2given∎

Depends on

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Sources

J. Lebl, Basic Analysis II, §8.5 (standard reference, not scraped)

The Jacobian sign of a regular C1C1 map is constant on a connected domain