Theorem

Sard's Theorem

Topology

The set of critical values of a sufficiently smooth function between manifolds has measure zero, meaning almost every value is a regular value. Proved by Arthur Sard, it is a foundational tool of differential topology, underlying transversality arguments and degree theory.

Facts

Statement

The set of critical values of a sufficiently smooth function between Euclidean spaces or manifolds has Lebesgue measure zero, so almost every point in the target is a regular value. 1

Proof Year

1942 1

The one-dimensional special case (target dimension m=1) was proven by Anthony P. Morse in 1939; Arthur Sard proved the general case in 1942.

Classification

Statement Form

Characterization Theorem 1

Connections

Has Statement Form

Characterization

In Branch

Topology

Sources

1. Sard's Theorem (Wikipedia)

Wikimedia Foundation

Statement section
...was proven by Anthony P. Morse in 1939, and the general case by Arthur Sard in 1942.
In Group: Geometric and Differential Topology, Lead
the set of critical values (that is, the image of the set of critical points) of a smooth function f from one Euclidean space or manifold to another is a null set

Switch Atlas

The Inner World

The Civil World

The Natural World

Sard's Theorem

Has Statement Form

In Branch