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.

Depends on

Used by

A degenerate critical set can be nonisolated Counterexample
An isolated critical point can be degenerate Counterexample
Morse functions and excellent Morse functions Definition
The intrinsic Hessian of a smooth function at a critical point Definition
The height function on the sphere is Morse and excellent Example
The zero-dimensional Morse convention Remark

Dependency tree · two levels

4 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

Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)
Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)