DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Critical points and critical values of a smooth function

Definition

Let $f : M \to R$ be a smooth function and let $p \in M$ .

The point $p$ is a critical point of $f$ when $d f_{p} = 0$ (The differential of a smooth real-valued function).
A real number $c$ is a critical value of $f$ when $c = f (p)$ for some critical point $p$ .

This is the specialization of Regular and critical points and values to real-valued functions: for a map to $R$ , the empty-fibre case is still regular, so a value is critical exactly when it is attained at a critical point.

Critical points and critical values of a smooth function

Definition

Depends on

Dependency tree · 2 levels

Sources