Sard's theorem

Remarks

When $n < m$, every point is critical, and the theorem says that the image $f(U)$ of a $C^1$ map is a null set. In the case $n = m$, the idea is this. Near a critical point, $f$ squeezes a small cube of side $r$ into a set of volume $o(r^n)$, because $Df(x)$ is singular. Covering the critical set by small cubes then gives total volume as small as one wants. The general case uses induction on $n$ and needs the higher smoothness to control the Taylor expansion along the critical set. Example: $f(x) = x^2$ on $\R$ has the single critical point $0$ and the single critical value $0$. The smoothness hypothesis cannot be dropped. Hassler Whitney constructed a $C^1$ function on $\R^2$ that is not constant on a connected arc of critical points, so its critical values fill an interval.

Going deeper

For a student

The complement of a null set is dense, so regular values are dense in $\R^m$. If $y$ is a regular value and $f^{-1}(y)$ is not empty, the implicit function theorem shows that $f^{-1}(y)$ is a submanifold of dimension $n - m$. This makes Sard's theorem a basic tool of differential topology, for example in transversality arguments and in the definition of the degree of a map. The same statement holds for smooth maps between manifolds, by working in charts.