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  "body": "\n## Example\n\nFor $x \\in \\R$, let $s(x)$ be the distance from $x$ to the nearest integer, so $0 \\le s(x) \\le 1/2$.\nThe *Takagi function* is\n$$T(x) = \\sum_{n=0}^{\\infty} \\frac{s(2^n x)}{2^n}.$$\nIts graph is the blancmange curve.\n\nEach term is continuous and bounded by $2^{-n-1}$.\nBy the [[Weierstrass M-test|0NX5NWAFC9]], the series converges uniformly on $\\R$, so $T$ is [[continuous|0P27W9RW0G]].\nThe function $T$ has period $1$ and satisfies $T(0) = 0$ and $T(1/2) = 1/2$.\nAt $x = 1/3$, each $2^n/3$ has distance $1/3$ to the nearest integer, so $T(1/3) = \\sum_{n \\ge 0} \\frac{1}{3 \\cdot 2^n} = \\frac 23$.\nThe maximum value of $T$ is $2/3$.\n\n## Remarks\n\nThe function $T$ is [[differentiable|0A61B60B5R]] at no point of $\\R$.\nThe idea of the proof: the partial sum $T_N = \\sum_{n \u003c N} s(2^n x)/2^n$ is linear on each interval $[k 2^{-N}, (k+1) 2^{-N}]$.\nIts slope there is an integer that changes by $\\pm 1$ from $T_N$ to $T_{N+1}$.\nSo the slopes of $T$ on small dyadic intervals around a point cannot converge.\nIt is a simpler companion of the [[Weierstrass function|01A0MK1730]], built from tent functions instead of cosines.\n\n## Explanations\n\n### L1 human\n\nStart with a tent of height $1/2$ over each unit interval.\nAdd tents half as tall and half as wide, then tents a quarter as tall and a quarter as wide, and so on forever.\nThe sum is a continuous curve that looks like a pudding with ripples at every scale.\nBecause it has corners at every scale, it has no tangent line anywhere.\nIt is a concrete continuous nowhere differentiable function.\n",
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