Sard's lemma

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Sard's lemma, also known as Sard's theorem or the Morse-Sard theorem, is a result in mathematical analysis which characterises the image of critical points of smooth functions $f$ from one Euclidean space to another as having Lebesgue measure 0. This makes it "small" in a sense. More precisely, if

$f:\mathbb{R}^n \rightarrow \mathbb{R}^m$

is smooth, and $C$ is the critical set of $f$ (the set in $\mathbb{R}^n$ of points $\mathbf{x}$ at which the Jacobian matrix of $f$ has rank $< m$ ), then $f (C)$ has measure 0 under the usual measure on $\mathbb{R}^m$ .

Thus $C$ can be large, but its image is small.

There are many variants of this lemma, which plays a basic role in singularity theory among other fields. The case $m = 1$ was proven by A. P. Morse in 1939, and the general case by Arthur Sard in 1942.

A version for infinite-dimensional Banach manifolds was proven by Stephen Smale.