# Sard's theorem

From Diffgeom

*This article gives the statement and possibly proof of a theorem that discusses regular values, critical values, regular points or critical points of a smooth map between differential manifolds*

## Statement

Suppose and are differential manifolds and is a smooth map between them. Then, the set of regular values of is a subset of measure zero in .

## Applications

Suppose and are differential manifolds, and the dimension of is strictly less than the dimension of . Then, if is a smooth map, the image has measure zero as a subset of . In particular, cannot be surjective.

This also shows that a differential manifold cannot be expressed as a union of the images of countably many smooth maps from differential manifolds of strictly smaller dimension