DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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The regular locus of a square-dimensional $C^{1}$ map

Definition

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, and let $f : U \to R^{n}$ be $C^{1}$ in the sense of Continuously differentiable maps, local inverses, and local diffeomorphisms. A point $a \in U$ is regular when the linear map $D f (a)$ is invertible (Invertible Euclidean linear maps). The regular locus is

$Reg (f) : = {a \in U : D f (a) is invertible} .$

Its complement in $U$ is the singular locus of $f$ .

Depends on

Used by

Dependency tree · two levels

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

The regular locus of a square-dimensional C1C1 map