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

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An injective regular $C^{1}$ map is a diffeomorphism onto its image

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, and let $f : U \to R^{n}$ be injective and $C^{1}$ . Suppose $D f (x)$ is invertible for every $x \in U$ , equivalently $U = Reg (f)$ (The regular locus of a square-dimensional $C^{1}$ map). Then $f [U]$ is open and $f : U \to f [U]$ is a $C^{1}$ diffeomorphism. Its inverse $g$ satisfies

$D g (f (x)) = D f (x)^{- 1} (x \in U) .$

Facts & Assumptions

Given: The hypotheses in the Statement and the definitions of a local inverse and diffeomorphism (Continuously differentiable maps, local inverses, and local diffeomorphisms, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

[L1]

Let $f : U \to R^{n}$ be $C^{1}$ and suppose $D f (x)$ is invertible for every $x \in U$ . Then $f$ maps every open subset of $U$ to an open subset of $R^{n}$ (A $C^{1}$ map with everywhere-invertible derivative is open).

[L2]

For the local inverse $g$ supplied by the inverse function theorem, $D g (y) = D f (g (y))^{- 1} (y \in W) .$ (The Euclidean inverse function theorem)

Proof

technique · direct

1.1

By [L1], $f [U]$ is open and the continuous bijection $f : U \to f [U]$ is open. Therefore its inverse $g : f [U] \to U$ is continuous.

L1given

2.1

Fix $y \in f [U]$ . The unique global inverse agrees near $y$ , by injectivity, with the local inverse from [L2]. Hence $g$ is $C^{1}$ near every image point and satisfies $D g (y) = D f (g (y))^{- 1}$ there. This proves the stated global $C^{1}$ diffeomorphism and derivative formula.

step 1.1L2given∎

Depends on

Used by

Dependency tree · two levels

21 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

University of Toronto MAT237, §3.3 (standard reference, not scraped)
J. Lebl, Basic Analysis II, §8.5 (standard reference, not scraped)

An injective regular C1C1 map is a diffeomorphism onto its image