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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be a connected open set and let $f \\colon U \\to \\mathbb{R}^m$ be [[differentiable|0135XZZQ8M]] on $U$.\nSuppose that $Df(x) = 0$ for each $x \\in U$.\nThen $f$ is constant on $U$.\n\n## Remarks\n\nThe proof fixes $a \\in U$ and shows that the set $S = \\{x \\in U : f(x) = f(a)\\}$ is open and closed in $U$. It is open by the [[mean value inequality|02CYB184RC]] with $M = 0$ on small balls, and closed because $f$ is continuous. As $U$ is connected and $a \\in S$, $S = U$.\nConnectedness is needed: on $U = (0, 1) \\cup (2, 3)$, the function that is $0$ on the first interval and $1$ on the second has derivative $0$ but is not constant.\nFor real-valued $f$, the condition $Df = 0$ means that all partial derivatives of $f$ vanish on $U$.\n\n## Explanations\n\n### L1 human\n\nSuppose that $f$ does not change in any direction at any point, and that the domain is in one piece. Then $f$ cannot change at all.\nOn a domain in two pieces, $f$ can take a different constant value on each piece.\n",
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