{
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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be differentiable on $[a, b]$, with [[one-sided derivatives|0HXEX0DPRA]] at the endpoints.\nLet $y$ be a real number strictly between $f'(a)$ and $f'(b)$.\nThen there is $c \\in (a, b)$ with $f'(c) = y$.\n\n## Remarks\n\nAssume $f'(a) \u003c y \u003c f'(b)$. The function $g(x) = f(x) - y x$ has $g'(a) \u003c 0 \u003c g'(b)$. So its minimum on $[a, b]$, given by the [[extreme value theorem|0WS1A8E2P9]], is at an interior point $c$. [[Fermat's theorem|0PCVBRV9VJ]] gives $g'(c) = 0$.\nSo a derivative has the intermediate value property, even when it is not continuous.\nIn particular, a derivative has no jump discontinuities. Mathlib states it as `exists_hasDerivWithinAt_eq_of_gt_of_lt`.\n\n## Explanations\n\n### L1 human\n\nThe derivative of a function cannot skip values, just as a continuous function cannot.\nFor example, no function has derivative $0$ on the left of a point and $1$ on the right.\n",
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    "name": "Darboux's theorem",
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  "name": "Darboux's theorem",
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