Weierstrass, Karl Wilhelm Theodor (1815–1897)

Karl Weierstrass was a German mathematician who is considered the father of modern analysis. Compelled by his father to study law, he instead spent four years at the University of Bonn, fencing, drinking, and reading math. He left under a cloud and ended up teaching in secondary schools for many years. In 1854 he published a paper, written 14 years earlier when he was fresh our of college, in Crelle's Journal, on Abelian functions which completed work that Niels Abel and Karl Jacobi had begun. Its importance was immediately recognized and Weierstrass was appointed a professor at the Royal Polytechnic School and a lecturer at the University of Berlin. He went on to give the first rigorous definitions of limit, derivative, differentiability, and convergence, and investigated under what conditions a power series will converge.


Karl Weierstrass


Weierstrass nondifferentiable function

Weierstrass nondifferentiable function is the earliest known example of a pathological function – a function that gives rise to a pathological curve. It was investigated by Weierstrass but had been first discovered by Bernhard Riemann, and is defined as:


    Weierstrass function


where 0 < a < 1, b is a positive odd integer, and ab > 1 + 3π/2.


The Weierstrass function is everywhere continuous but nowhere differentiable; in other words, no tangent exists to its curve at any point. Constructed from an infinite sum of trigonometric functions, it is the densely-nested oscillating structure that makes the definition of a tangent line impossible.