Weierstrass Function

In 1872, Karl Weierstrass shocked the mathematical world by exhibiting a function that is continuous at every point yet differentiable at none. This contradicted the widely held intuition that continuous curves must be smooth "almost everywhere."

The construction is a sum of cosines whose frequencies grow geometrically while amplitudes decay. When a·b > 1 + 3π/2, the function is provably nowhere differentiable. Zoom in as far as you like — the jaggedness never resolves into smoothness.

The fractal dimension of the graph is 2 − log(a)/log(b), nestled between a line (dim 1) and a plane (dim 2).