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  "body": "\n## Summary\n\nThe smoothness classes $C^0, C^1, C^2, \\ldots, C^\\infty$ sort functions by how many continuous derivatives they have.\nA function of class $C^k$ has $k$ derivatives, and the $k$-th one is continuous.\nThe classes are nested, each strictly inside the one before, and their intersection is the class $C^\\infty$ of [[smooth functions|0V33BNKZ9S]].\nTheorems in analysis and geometry state their hypotheses in terms of these classes.\n\n## Explanations\n\n### L1 human\n\nSome functions are continuous but have corners, like $\\abs{x}$.\nSome have a derivative, but the derivative has a corner, like $x \\abs{x}$, whose derivative is $2\\abs{x}$.\nCounting how many times you can differentiate before a corner shows up gives the smoothness class.\nSo $\\abs{x}$ is of class $C^0$, $x\\abs{x}$ is of class $C^1$, and a polynomial is of class $C^\\infty$.\n",
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