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  "body": "\n## Statement\n\nLet $n, m \\ge 1$, let $U \\subseteq \\R^n$ be open, and let $f \\colon U \\to \\R^m$ be of class $C^k$, where $k \\ge \\max(n - m + 1, 1)$.\nA point $x \\in U$ is a [[critical point|0YTCMFB4HC]] of $f$ if the [[Jacobian matrix|047NDTEPCV]] $Df(x)$ has rank less than $m$.\nA critical value of $f$ is the image $f(x)$ of a critical point $x$.\nThen the set of critical values of $f$ has [[Lebesgue measure|2Z2QVWZ262]] zero in $\\R^m$.\n\n## Remarks\n\nWhen $n \u003c m$, every point is critical, and the theorem says that the image $f(U)$ of a $C^1$ map is a [[null set|VQQM4Y550T]].\nIn the case $n = m$, the idea is this.\nNear a critical point, $f$ squeezes a small cube of side $r$ into a set of volume $o(r^n)$, because $Df(x)$ is singular.\nCovering the critical set by small cubes then gives total volume as small as one wants.\nThe general case uses induction on $n$ and needs the higher smoothness to control the Taylor expansion along the critical set.\nExample: $f(x) = x^2$ on $\\R$ has the single critical point $0$ and the single critical value $0$.\nThe smoothness hypothesis cannot be dropped.\nHassler 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.\n\n## Explanations\n\n### L1 human\n\nA critical value is a value that a map takes at a point where it flattens out.\nSard's theorem says that a smooth enough map has very few critical values: they form a set of measure zero.\nSo a value picked at random is almost surely a regular value.\n\n### L2 human\n\nThe complement of a null set is dense, so regular values are dense in $\\R^m$.\nIf $y$ 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 $n - m$.\nThis 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.\nThe same statement holds for smooth maps between manifolds, by working in charts.\n",
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