Karl Weierstrass

Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions.

Delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.

In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg. Besides mathematics he also taught physics, botany, and gymnastics.

After 1850 Weierstrass suffered from a long period of illness, but was able to publish papers that brought him fame and distinction. The University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut, which later became the Technical University of Berlin. In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became the Humboldt Universität zu Berlin.

At the age of fifty-five, Weierstrass met Sonya Kovalevsky whom he tutored privately after failing to secure her admission to the University. They had a fruitful intellectual, but troubled personal relationship that "far transcended the usual teacher-student relationship". The misinterpretation of this relationship and Kovalevsky's early death in 1891 was said to have contributed to Weierstrass' later ill-health. He was immobile for the last three years of his life, and died in Berlin from pneumonia.

f ( x ) {\displaystyle \displaystyle f(x)} is continuous at x = x 0 {\displaystyle \displaystyle x=x_{0}} if ∀   ε > 0   ∃   δ > 0 {\displaystyle \displaystyle \forall \ \varepsilon >0\ \exists \ \delta >0} such that for every x {\displaystyle x} in the domain of f {\displaystyle f} ,     | x − x 0 | < δ ⇒ | f ( x ) − f ( x 0 ) | < ε . {\displaystyle \displaystyle \ |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\varepsilon .} In simple English, f ( x ) {\displaystyle \displaystyle f(x)} is continuous at a point x = x 0 {\displaystyle \displaystyle x=x_{0}} if for each ε > 0   {\displaystyle \varepsilon >0\ } there exists a δ > 0 {\displaystyle \delta >0} such that the function f ( x ) {\displaystyle f(x)} lies between f ( x 0 ) − ε {\displaystyle f(x_{0})-\varepsilon } and f ( x 0 ) + ε {\displaystyle f(x_{0})+\varepsilon } when x {\displaystyle x} is between x 0 − δ {\displaystyle x_{0}-\delta } and x 0 + δ {\displaystyle x_{0}+\delta } .

Weierstrass also made significant advancements in the field of calculus of variations. Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory which paved the way for the modern study of the calculus of variations. Among the several significant axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition, which gives sufficient conditions for an extremal to have a corner along a given extrema, and allows one to find a minimizing curve for a given integral.