# 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.

## 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:

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.