Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the "father of modern analysis".

Soundness of calculus

Weierstrass was interested in the soundness of calculus. At the time, there were no unambiguous definitions regarding the fundaments of calculus, hence theorems could not be properly proven. 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 other eminent mathematicians such as Cauchy had only vague definitions of limits and continuity of functions. Weierstrass defined continuity as follows:

Related Topics:
Soundness - Bolzano - Limit - 1817 - Cauchy - Continuity

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f(x) is continous at x = x_0 if given that for an arbitrary arepsilon ge 0 there exists delta ge 0 such that

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|x-x_0| < delta implies |f(x) - f(x_0)| < arepsilon.

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Weierstrass also formulated the definition of limit and derivative still in use today.

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With these new definitions he was able to write proofs of several at the time unproven theorems such as the intermediate value theorem, Bolzano-Weierstrass theorem and Heine-Borel theorem.

Related Topics:
Intermediate value theorem - Bolzano-Weierstrass theorem - Heine-Borel theorem

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