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  "body": "\n## Counterexample\n\nLet $0 \u003c a \u003c 1$, and let $b$ be a positive odd integer with $a b \u003e 1 + \\frac{3}{2} \\pi$. The Weierstrass function is\n$$W(x) = \\sum_{n=0}^{\\infty} a^n \\cos(b^n \\pi x).$$\nThe function $W$ is continuous on $\\mathbb{R}$, but it is not [[differentiable|0A61B60B5R]] at any point.\nSo a continuous function need not have a derivative anywhere.\n\n## Remarks\n\nThe terms satisfy $|a^n \\cos(b^n \\pi x)| \\le a^n$. The [[M-test|0NX5NWAFC9]] and the [[continuity of uniform sums|0R0MYT0BJD]] give continuity.\nG. H. Hardy later showed that the conclusion holds under the weaker condition $a b \\ge 1$.\nKarl Weierstrass presented the function to the Prussian Academy of Sciences on 18 July 1872.\nBernard Bolzano had constructed such a function around 1831, but it was not published until 1922.\n\n## Explanations\n\n### L1 human\n\nThe graph of the Weierstrass function wiggles at every scale.\nZooming in never shows a straight piece, so there is no tangent line at any point.\n",
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