The 1872 curve that is continuous everywhere yet smooth nowhere
Mathematicians including Gauss had assumed a continuous curve could only have isolated sharp points. On 18 July 1872 Karl Weierstrass presented a function to the Royal Academy of Sciences in Berlin that is unbroken everywhere yet has no derivative at any point, zigzagging at every scale like a fractal.
It was the first published function built specifically to break that belief, and it overturned several proofs that leaned on geometric intuition and loose ideas of smoothness. Contemporaries disliked such monsters; Charles Hermite called one class of them a lamentable scourge. Weierstrass defined it as an infinite Fourier series of cosine waves with rapidly rising frequencies. Because the terms are bounded, the series converges uniformly, so the result is continuous, but its oscillations never settle.
Zoom in on any piece and it never straightens into a line; between any two points, however close, the function is not monotone. That detail at every level makes it one of the first fractals studied, long before the word existed. Pinning down the Hausdorff dimension of its graph remained an open problem until 2018. Such curves were hard to picture until computers arrived, and gained wide acceptance only when models of Brownian motion demanded infinitely jagged paths.
The frantic wiggling is essential. Tamer functions cannot pull off the trick: Rademacher's theorem says a Lipschitz function fails to be differentiable only on a set of measure zero, and the same holds for monotone functions. When people sketch a continuous curve, they almost always draw one of these well-behaved kinds, which may explain why the old assumption survived.
Weierstrass built on an earlier example from Bernhard Riemann, who insisted his function was nowhere differentiable but published no proof; Weierstrass could trace no proof in Riemann's papers or through his pupils. G. H. Hardy later established, in 1916, that it lacks a finite derivative at every irrational point, yet in 1969 Joseph Gerver showed it does have a derivative at certain rational multiples of pi, and he settled the question in 1971. Riemann's function turns out to be differentiable almost nowhere, which is not quite nowhere.
Source: Weierstrass function
