Sard's theorem

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 $M$ and $N$ are differential manifolds and $f:M\to N$ is a smooth map between them. Then, the set of regular values of $f$ is a subset of measure zero in $N$.

Applications

Suppose $M$ and $N$ are differential manifolds, and the dimension of $M$ is strictly less than the dimension of $N$. Then, if $f:M\to N$ is a smooth map, the image $f(M)$ has measure zero as a subset of $N$. In particular, $f$ 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