Weierstrass function (nowhere-differentiable function)

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In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve. The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness. These types of functions were disliked by contemporaries. For instance, Charles Hermite, on finding that one class of function he was working on had such a property, described it as a "lamentable scourge". The functions were difficult to visualize until the arrival of computers in the next century...

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