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  "body": "\n## Definition\n\nLet $I$ be an open interval and let $f \\colon I \\to \\mathbb{R}$.\nThe function $f$ is continuously differentiable, or of class $C^1$, if $f$ is [[differentiable|0A61B60B5R]] on $I$ and $f'$ is [[continuous|0P27W9RW0G]] on $I$.\nMore generally, for $k \\ge 1$, $f$ is of class $C^k$ if its [[higher derivative|084NGSTWM7]] $f^{(k)}$ exists and is continuous on $I$.\nThe class $C^0$ consists of the continuous functions.\n\n## Remarks\n\nEach $C^{k+1}$ function is $C^k$.\nA differentiable function need not be $C^1$. See [[x^2 sin(1/x)|0WDAKKN4NC]].\nThe function $x |x|$ is $C^1$ but not twice differentiable at $0$.\nMathlib defines the class as `ContDiff ℝ k f`.\n\n## Explanations\n\n### L1 human\n\nA $C^1$ function has a slope at each point, and the slope changes without jumps.\nThe class $C^k$ asks the same of the first $k$ derivatives.\n",
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