Karl Weierstrass (31 Oct 1815 – 19 Feb 1897) constructed what a function looks like that’s continuous everywhere but has no derivative at any point:
f(x) = ∑_{n=0}^∞ aⁿ cos(bⁿ π x).
The magnified inset reveals its fractal self-similarity at every scale.
“A mathematician who is not also something of a poet will never be a complete mathematician.”
Today this function underpins fractal geometry and models irregular real-world phenomena; from coastlines and turbulent flows in physics to stock-price volatility in finance and rough signals in engineering.