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  "body": "\n## Statement\n\nLet $n \\ge 1$ and $m \\ge 0$ be integers and let $A \\subseteq \\R^n$ be a [[closed set|0ZCK8DQYN1]].\nFor each [[multi-index|0PJXBR6QA3]] $\\alpha$ with $\\abs{\\alpha} \\le m$, let $f_\\alpha \\colon A \\to \\R$ be continuous.\nFor $x, y \\in A$ and $\\abs{\\alpha} \\le m$, set\n$$R_\\alpha(x, y) = f_\\alpha(x) - \\sum_{\\abs{\\beta} \\le m - \\abs{\\alpha}} \\frac{f_{\\alpha + \\beta}(y)}{\\beta!} (x - y)^\\beta.$$\nAssume that for every compact set $K \\subseteq A$ and every $\\abs{\\alpha} \\le m$,\n$$\\frac{\\abs{R_\\alpha(x, y)}}{\\norm{x - y}^{m - \\abs{\\alpha}}} \\to 0 \\quad \\text{as } \\norm{x - y} \\to 0,$$\nuniformly for $x, y \\in K$ with $x \\ne y$.\nThen there is a function $F \\colon \\R^n \\to \\R$ of [[class $C^m$|0TCBZMEYGN]] such that $D^\\alpha F = f_\\alpha$ on $A$ for every $\\abs{\\alpha} \\le m$.\nMoreover, $F$ can be chosen of class $C^\\infty$ on $\\R^n \\setminus A$.\n\n## Remarks\n\nThe family $(f_\\alpha)$ is a list of candidate derivatives of $F$ on $A$, called a jet.\nThe condition says that the candidate values obey [[Taylor's theorem|0J8T4K4YTE]] uniformly on compact parts of $A$.\nIt is necessary: if $F$ is of class $C^m$ on $\\R^n$, then the restrictions of $D^\\alpha F$ to $A$ satisfy it, by Taylor's theorem.\nFor $m = 0$ the condition holds automatically, because a continuous function is uniformly continuous on compact sets, and the theorem gives a continuous extension, as the Tietze extension theorem does.\nThe proof covers $\\R^n \\setminus A$ by cubes whose size is comparable to their distance to $A$.\nIt then glues Taylor polynomials taken at nearby points of $A$ with a partition of unity subordinate to these cubes.\n\n## Explanations\n\n### L1 human\n\nSuppose the values of a function and of its derivatives are prescribed only on a closed set.\nThe question is whether a smooth function on all of space has exactly these values and derivatives there.\nWhitney's theorem says yes, as soon as the prescribed values fit together the way Taylor expansions of a smooth function would.\n\n### L2 human\n\nThe theorem turns questions about functions on arbitrary closed sets into questions about jets, which are easier to handle.\nIt is used to define smooth functions on closed subsets, and in the study of ideals of smooth functions.\nFor $n = 1$ and $m = 1$ the condition says that $f_0$ is differentiable along $A$ with derivative $f_1$, uniformly on compact sets.\nThe related [[Borel's lemma|KZHNA19FSS]] extends a jet given at a single point.\n",
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