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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $f \\colon U \\to \\mathbb{R}^m$ be of [[class C^1|0TCBZMEYGN]] on $U$.\nThen $f$ is [[strictly differentiable|0146MBQNNE]] at each point $a \\in U$, with linear map $Df(a)$.\n\n## Remarks\n\nProof: let $\\varepsilon \u003e 0$ and choose $r \u003e 0$ with $\\|Df(z) - Df(a)\\| \\le \\varepsilon$ for $\\|z - a\\| \u003c r$. Apply the [[mean value inequality|02CYB184RC]] to $g(z) = f(z) - Df(a) z$ on the ball of radius $r$. This gives $\\|f(x) - f(y) - Df(a)(x - y)\\| \\le \\varepsilon \\|x - y\\|$ for $x, y$ in that ball.\nFor a function of one variable on an open interval, strict differentiability at each point is equivalent to being of class $C^1$.\n\n## Explanations\n\n### L1 human\n\nIf the derivative changes continuously, then near $a$ it stays close to $Df(a)$.\nSo every chord of the graph near $a$ has a slope close to $Df(a)$, which is what strict differentiability asks.\n",
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