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  "body": "\n## Counterexample\n\nLet $\\varphi \\colon \\mathbb{R} \\to \\mathbb{R}$ be the zigzag function with $\\varphi(x) = |x|$ for $-1 \\le x \\le 1$ and $\\varphi(x + 2) = \\varphi(x)$ for all $x$.\nDefine\n$$f(x) = \\sum_{n=0}^{\\infty} \\Big( \\frac{3}{4} \\Big)^n \\varphi(4^n x).$$\nThe function $f$ is continuous on $\\mathbb{R}$, but it is not [[differentiable|0A61B60B5R]] at any point.\n\n## Remarks\n\nSince $0 \\le \\varphi \\le 1$, the [[M-test|0NX5NWAFC9]] with $M_n = (3/4)^n$ and the [[continuity of uniform sums|0R0MYT0BJD]] give continuity.\nFix $x$. For each $m$, choose $\\delta_m = \\pm \\frac{1}{2} 4^{-m}$ so that no integer lies strictly between $4^m x$ and $4^m (x + \\delta_m)$.\nThen $\\Big| \\frac{f(x + \\delta_m) - f(x)}{\\delta_m} \\Big| \\ge \\frac{3^m + 1}{2}$. This tends to infinity while $\\delta_m \\to 0$, so $f'(x)$ does not exist.\nThis function replaces the cosine in the [[Weierstrass function|01A0MK1730]] by a piecewise linear zigzag.\n\n## Explanations\n\n### L1 human\n\nEach term adds smaller and steeper zigzags to the graph.\nThe sum is continuous, but its graph has a corner at every scale near every point.\n",
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