Karl Weierstrass

Karl Weierstrass (1815–1897) is called the father of modern analysis, and he earned the title by being the most careful man in the history of the subject. Cauchy had begun making calculus rigorous; Weierstrass finished the job, banishing vague talk of quantities "approaching" or "becoming infinitely small" and replacing it with the cold, exact ε–δ language you still learn today.

\forall \varepsilon > 0 \;\; \exists \delta > 0 : \; 0 < |x - a| < \delta \;\Rightarrow\; |f(x) - L| < \varepsilon.

The schoolteacher who came from nowhere

Here's the unlikely part: Weierstrass spent his thirties as an obscure secondary-school teacher in provincial Germany, teaching not just maths but handwriting, geography and gymnastics, and doing original research alone at night with no library and no colleagues. Then, in his early forties, he published one paper — and the mathematical world discovered a giant had been hiding among the schoolmasters. He was handed a university chair almost overnight.

The monster with no tangents

His most shocking gift was a warning. Everyone had assumed a continuous curve must be smooth almost everywhere — you could surely draw a tangent at most points. Weierstrass built a function that is continuous everywhere yet differentiable nowhere: unbroken, but infinitely jagged at every single point, with no tangent anywhere at all.

W(x) = \sum_{n=0}^{\infty} a^{n} \cos\!\big(b^{n} \pi x\big), \qquad 0 < a < 1.

Contemporaries recoiled — one called such things "a deplorable evil." Today you'd recognise its wild, self-similar roughness instantly: it is a fractal, and the same nowhere-smooth behaviour drives the Brownian motion that prices options.

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The full story (with far fewer jokes) is on Wikipedia: Karl Weierstrass — Wikipedia.