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  "body": "\n## Example\n\nLet $h$ be a function that is differentiable at every point.\nIts [[derivative|0V1EJCT8G6]] $h'$ is a Pompeiu derivative if $h'$ vanishes on a [[dense set|0BSJSX7ME2]] but is not identically zero.\nHere is a construction.\nLet $q_1, q_2, \\dots$ list the rational numbers in $[0, 1]$, and write $u^{1/3}$ for the real cube root.\nFor $x \\in [0, 1]$ set\n$$g(x) = \\sum_{n=1}^\\infty 2^{-n} (x - q_n)^{1/3}.$$\nThe series converges uniformly, and each term is increasing, so $g$ is continuous and [[strictly increasing|011EJQWWJS]].\n\nEach difference quotient of $u \\mapsto u^{1/3}$ at a point $u \\ne 0$ is positive and at most $\\frac{4}{3} \\abs{u}^{-2/3}$, since $a^2 + ab + b^2 \\ge \\frac{3}{4} a^2$ for real $a, b$.\nUse this bound with dominated convergence where the series below is finite, and Fatou's lemma where it is infinite.\nIt follows that $g$ has at each $x$ a derivative in $(0, \\infty]$, namely\n$$g'(x) = \\sum_{n=1}^\\infty \\frac{2^{-n}}{3} \\abs{x - q_n}^{-2/3},$$\nwith $g'(q_n) = \\infty$ for every $n$ (one-sided at the end points).\nSince $\\abs{x - q_n} \\le 1$, every term is at least $2^{-n}/3$, so $g'(x) \\ge 1/3$.\n\nLet $h = g^{-1}$ on the interval $[g(0), g(1)]$.\nBy the rule for the derivative of an inverse function, $h$ is differentiable everywhere, with $h'(y) = 1/g'(h(y))$, read as $0$ when $g'(h(y)) = \\infty$.\nSo $0 \\le h' \\le 3$.\nThe derivative $h'$ vanishes at every point $g(q_n)$, and these points are dense in $[g(0), g(1)]$ because $g$ is a homeomorphism.\nBut $h'$ is not identically zero, since $h$ is strictly increasing and a function with zero derivative is constant.\nSo $h'$ is a Pompeiu derivative.\n\n## Explanations\n\n### L1 human\n\nOne might think that a derivative which is zero on a dense set of points must be zero everywhere.\nThis example shows that this is false.\nThe function $g$ has vertical tangents at all rational points, so its inverse $h$ has horizontal tangents at a dense set of points, and yet $h$ keeps increasing.\nLike every derivative, $h'$ still has the intermediate value property, so it is a [[Darboux function|7BJEH1GCTE]].\n",
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