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The function is an infinite sum, starting with k=0. The kth term is a^{k} cos(b^{k} π x), where a is a number between 0 and 1, b is an odd integer, and the product ab is greater than 1 + 3π/2. The LaTeX code for the formula as displayed is: \begin{gather} f(x) = \sum_{k=0}^{\infty} a^{k} \cos\Big(b^k \pi x\Big) \\ 0 < a < 1, \qquad b \in \mathbb{Z}_\text{odd}, \qquad a\,b > 1 + \frac{3\pi}{2} \end{gather*}\\

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The function is an infinite sum, starting with k=0. The kth term is a^{k} cos(b^{k} π x), where a is a number between 0 and 1, b is an odd integer, and the product a*b is greater than 1 + 3π/2. The LaTeX code for the formula as displayed is: \begin{gather*} f(x) = \sum_{k=0}^{\infty} a^{k} \cos\Big(b^k \pi x\Big) \\ 0 < a < 1, \qquad b \in \mathbb{Z}_\text{odd}, \qquad a\,b > 1 + \frac{3\pi}{2} \end{gather*}\\
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Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 13:31:06 UTC Robert McNees
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Henri Poincare was the first to understand the terrible thing Weierstrass had done. He called it "an outrage against common sense.”
Somehow, Weierstrass had stitched together a pathological function made out of pointed teeth and sharp corners, and brought it to life.
"Weierstrass's Monster," as Poincare called it, is a function that is [lightning flashes outside the window] everywhere continuous but nowhere differentiable [huge crash of thunder].

In conversation about a year ago from mastodon.social permalink