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  "body": "\n## Counterexample\n\nLet $f(x) = x^2 \\sin(1/x)$ for $x \\ne 0$, and let $f(0) = 0$.\nThen $f$ is [[differentiable|0A61B60B5R]] on $\\mathbb{R}$, but $f'$ is not continuous at $0$.\nAt $0$, the difference quotient is $h \\sin(1/h)$, which tends to $0$. So $f'(0) = 0$.\nFor $x \\ne 0$, $f'(x) = 2 x \\sin(1/x) - \\cos(1/x)$. This has no limit as $x \\to 0$, because $\\cos(1/x)$ has none.\n\n## Remarks\n\nSo $f$ is differentiable but not [[of class $C^1$|06SKCHDQT5]].\nThe discontinuity of $f'$ at $0$ is essential. By [[Darboux's theorem|0Q5A47J48D]], a derivative can have no jump discontinuities.\n\n## Explanations\n\n### L1 human\n\nThe factor $x^2$ flattens the oscillation of $\\sin(1/x)$ enough to give a slope $0$ at the origin.\nBut near the origin the slopes keep swinging between about $-1$ and $1$.\n",
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  "msc": [
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  "name": "x^2 sin(1/x) is differentiable with discontinuous derivative",
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