Weierstrass, who was born at Ostenfelde in Germany, spent four years at the University of Bonn studying law to please his father. After abandoning law he trained as a school teacher and spent nearly 15 years teaching at elementary schools in obscure German villages. However, he found time to combine his mathematical researches with his school teaching and in 1854 he attracted considerable favorable attention with a memoir on Abelian functions, which he published in Crelle's journal. The fame this work brought him resulted in his obtaining a post as professor of mathematics at the Royal Polytechnic School in Berlin and he soon moved on to the University of Berlin.

Weierstrass's work on Abelian functions is generally considered to be his finest, but he made numerous other contributions to many other areas of mathematics. He was one of the first to make systematic use in analysis of representations of functions by power series. He was a superb and very influential teacher, an excellent fencer, and, unlike many mathematicians, he intensely disliked music. His work in ‘arithmetizing’ analysis led him into a fierce controversy with the constructivist Leopold Kronecker, who thought that Weierstrass's widespread use of nonconstructive proofs and definitions was unsound.

It is to Weierstrass together with Augustin Cauchy that modern analysis is indebted for its high standards of rigor. Weierstrass gave the first truly rigorous definitions of such fundamental analytical concepts as limit, continuity, differentiability, and convergence. He also did very important work in investigating the precise conditions under which infinite series converged. Tests for convergence that he devised are still in use.