One of a family of theorems that

smooth mappings have relatively few

critical points. The usual form is, if f is a C

^{1} mapping of smooth

manifolds X to Y, the image under f of the set of

critical points of f has

Lebesgue measure zero in Y. You can find this form in most books on

manifolds, for example

Calculus on manifolds by

Michael Spivak.

A more precise form of this theorem is due to Herbert Federer:

Suppose f: U → Y is a map of class C^{k}, from an open set U ⊂ **R**^{m} into any normed space Y. Let B ⊂ U be the set of points x ∈ U where the derivative Df(x) has rank ≤ n, where n < m is fixed. Then the image f(B) has s-dimensional Hausdorff measure zero, where s = n + (m - n)/k.

Thus, for a C^{1} map, the best you can do is the classical Sard's theorem where s = m (**H**^{m} is not exactly the Lebesgue measure of the classical theorem, but close enough). With more smoothness you get lower dimensions, down to a minimum of n. The bound given for s is sharp: Federer constructed examples of maps with **H**^{t}(B) > 0 for any t < n + (m-n)/k.

The proof of this sharp form of Sard's theorem is quite hard; it depends on rather technical methods of geometric measure theory. See *Geometric measure theory* by Herbert Federer, section 3.4.3.

If the domain of f is instead a Banach space, there is a form called the Morse-Sard theorem which states that f(B) is of first Baire category in Y. This is much coarser, but it's the best you can do in infinite dimensions.

This theorem is an example of a big rock, because it is frequently applied in very "geometric" contexts in differential topology and differential geometry, where people may not be very concerned with the hard analytic details that come into the proof. In differential topology all you want to know is that maps with no critical points in the "wrong places" are dense; see transversality for more information.