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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}^m$, and let $k \\ge 1$ be an integer.\nSay that $f$ is of class $C^k$ on $U$ when the following is true for each component $f_j$ and each $r \\le k$. Each [[partial derivative|02RSXVM1ZN]] $\\partial^r f_j / \\partial x_{i_r} \\cdots \\partial x_{i_1}$ of order $r$ exists on $U$ and is continuous on $U$.\nThen $f$ is of class $C^k$ on $U$ if and only if these two conditions hold.\n\n1. For each $r$ with $0 \\le r \\le k$, the [[iterated derivative|0EY05Z9RPN]] $D^r f$ exists on $U$ and is continuous on $U$.\n2. For each $r$ with $0 \\le r \u003c k$, the map $D^r f$ is differentiable on $U$.\n\n## Remarks\n\nSo in finite dimensions, $C^k$ means $k$ times continuously differentiable, in either the sense of partial derivatives or the sense of Fréchet derivatives.\nA function is of class $C^\\infty$, or smooth, when it is of class $C^k$ for each $k$.\nEach polynomial in $x_1, \\dots, x_n$ is of class $C^\\infty$ on $\\mathbb{R}^n$. The function $x \\mapsto \\|x\\|$ is of class $C^\\infty$ on $\\mathbb{R}^n \\setminus \\{0\\}$, but it is not differentiable at $0$.\n\n## Explanations\n\n### L1 human\n\nThere are two ways to count derivatives: one variable at a time with partial derivatives, or all variables at once with total derivatives.\nFor functions on open subsets of $\\mathbb{R}^n$, both ways give the same classes $C^1, C^2, \\dots$, as long as you also ask for continuity.\n",
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