Birthday of Karl Weierstrass (1815)

Called the 'father of modern analysis,' Weierstrass gave the rigorous ε-δ definition of a limit, taming the intuitive but imprecise reasoning that had plagued calculus for 200 years. More startlingly, he exhibited a function that is continuous everywhere but differentiable nowhere — a curve with no tangent at any point: W(x) = Σ aⁿ cos(bⁿπx). This violated everyone's geometric intuition and showed that smooth-looking curves could be pathologically irregular. He also proved that any continuous function on a closed interval can be approximated arbitrarily closely by polynomials.