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.[6][7] 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.

The formal definition of continuity of a function, as formulated by Weierstrass, is as follows:

${\displaystyle \displaystyle f(x)}$ is continuous at ${\displaystyle \displaystyle x=x_{0}}$ if ${\displaystyle \displaystyle \forall \ \varepsilon >0\ \exists \ \delta >0}$ such that for every ${\displaystyle x}$ in the domain of ${\displaystyle f}$,   ${\displaystyle \displaystyle \ |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\varepsilon .}$ In simple English, ${\displaystyle \displaystyle f(x)}$ is continuous at a point ${\displaystyle \displaystyle x=x_{0}}$ if for each ${\displaystyle x}$ close enough to ${\displaystyle x_{0}}$, the function value ${\displaystyle f(x)}$ is very close to ${\displaystyle f(x_{0})}$, where the "close enough" restriction typically depends on the desired closeness of ${\displaystyle f(x_{0})}$ to ${\displaystyle f(x).}$ Using this definition, he proved the Intermediate Value Theorem. He also proved the Bolzano–Weierstrass theorem and used it to study the properties of continuous functions on closed and bounded intervals.

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.

• Stone–Weierstrass theorem
• Casorati–Weierstrass–Sokhotski theorem
• Weierstrass's elliptic functions
• Weierstrass function
• Weierstrass M-test
• Weierstrass preparation theorem
• Lindemann–Weierstrass theorem
• Weierstrass factorization theorem
• Enneper–Weierstrass parameterization