Weierstrass FunctionA curve that is perfectly continuous — no jumps, no breaks — and yet has no slope anywhere at all. Zoom into any point as far as you like: instead of flattening into a straight line the way a smooth curve would, it keeps throwing out fresh wiggles at every scale, forever.Karl Weierstrass built it in 1872, adding ever finer cosine ripples aⁿcos(bⁿπx) with a<1 shrinking each ripple's height while b>1 multiplies its frequency. Whenever the product a·b exceeds 1, the frequency wins the race against the shrinkage and the roughness never gets ironed out — the first rigorous example of a function continuous everywhere but differentiable nowhere, a monster that broke 19th-century intuition about what a curve could be.