Editing Sard's theorem (section)

== Variants ==
There are many variants of this lemma, which plays a basic role in [[singularity theory]] among other fields. The case <math>m=1</math> was proven by [[Anthony P. Morse]] in 1939,<ref>{{citation  | first= Anthony P.  | last=Morse   | author-link = Anthony Morse   | title=The behaviour of a function on its critical set  | journal=[[Annals of Mathematics]]  | volume=40  | issue=1  |date=January 1939  | pages=62–70  | jstor=1968544  | doi=10.2307/1968544  | bibcode=1939AnMat..40...62M | mr=1503449  |postscript=.}}</ref> and the general case by [[Arthur Sard]] in 1942.<ref name="Sard1942" />

A version for infinite-dimensional [[Banach manifold]]s was proven by [[Stephen Smale]].<ref>{{citation  | first=Stephen  | last=Smale  | author-link=Stephen Smale  | title=An Infinite Dimensional Version of Sard's Theorem  | journal=[[American Journal of Mathematics]] | volume=87   | year=1965  | pages=861–866  | jstor= 2373250  | doi=10.2307/2373250  | issue=4  | mr=0185604   | zbl=0143.35301 |postscript=. }}</ref>

The statement is quite powerful, and the proof involves analysis. In [[topology]] it is often quoted — as in the [[Brouwer fixed-point theorem]] and some applications in [[Morse theory]] — in order to prove the weaker corollary that “a non-constant smooth map has '''at least one''' regular value”.

In 1965 Sard further generalized his theorem to state that if <math>f:N\rightarrow M</math> is <math>C^\infty</math> and if <math>A_r\subseteq N</math> is the set of points <math>x\in N</math> such that <math>df_x</math> has rank less or equal than <math>r</math>, then the [[Hausdorff dimension]] of <math>f(A_r)</math> is at most <math>r</math>.<ref>{{citation  | first=Arthur  | last=Sard  | title=Hausdorff Measure of Critical Images on Banach Manifolds  | journal=[[American Journal of Mathematics]]  | volume=87  | year=1965  | pages=158–174  | doi=10.2307/2373229  | issue=1  | jstor=2373229  | mr=0173748  | zbl=0137.42501 }} and also {{Citation  | first=Arthur  | last=Sard  | title = Errata to ''Hausdorff measures of critical images on Banach manifolds''  | journal=[[American Journal of Mathematics]]  | volume=87  | year=1965  | pages=158–174  | issue=3  | jstor = 2373074  | doi = 10.2307/2373229  | mr = 0180649  | zbl = 0137.42501 |postscript=. }}</ref><ref>{{citation |title=Show that <var>f(C)</var> has Hausdorff dimension at most zero |date=July 18, 2013 |work=[[Stack Exchange]] |url=https://math.stackexchange.com/q/446049 }}</ref>