Editing Karl Weierstrass (section)

== Mathematical contributions ==

=== Soundness of calculus ===
Weierstrass was interested in the [[soundness]] of calculus, and at the time there were somewhat ambiguous definitions of the foundations of calculus so that important theorems could not be proven with sufficient rigour.  Although [[Bernard Bolzano|Bolzano]] had developed a reasonably rigorous definition of a [[Limit of a function|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 [[Limit of a function|limits]] and [[Continuous function|continuity]] of functions.

The basic idea behind [[(ε, δ)-definition of limit|Delta-epsilon]] proofs is, arguably, first found in the works of [[Augustin-Louis Cauchy|Cauchy]] in the 1820s.<ref>{{citation
|title=Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
|first=Judith V.
|last=Grabiner
|journal=The American Mathematical Monthly
|date=March 1983
|volume=90
|pages=185–194
|url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Grabiner185-194.pdf |archive-url=https://web.archive.org/web/20141129124944/http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Grabiner185-194.pdf |archive-date=2014-11-29 |url-status=live
|doi=10.2307/2975545
|issue=3
|jstor=2975545
}}</ref><ref>{{citation
 |first        = A.-L.
 |last         = Cauchy
 |author-link  = Augustin-Louis Cauchy
 |title        = Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal
 |place        = Paris
 |year         = 1823
 |url          = http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
 |chapter      = Septième Leçon – Valeurs de quelques expressions qui se présentent sous les formes indéterminées <math>\frac{\infty}\infty, \infty^0, \ldots</math> Relation qui existe entre le rapport aux différences finies et la fonction dérivée
 |chapter-url   = http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f45n5.capture
 |page   = [http://gallica.bnf.fr/ark:/12148/bpt6k90196z.image.f47 44]
 |access-date  = 2009-05-01
 |archive-url  = https://www.webcitation.org/5gVUmywgY?url=http://math-doc.ujf-grenoble.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_4_9_0
 |archive-date = 2009-05-04
 |url-status     = dead
}}</ref>
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 that is false in general. The correct statement is rather that the [[uniform limit|''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:

<math>\displaystyle f(x)</math> is continuous at <math>\displaystyle x = x_0</math> if <math> \displaystyle \forall \ \varepsilon > 0\ \exists\ \delta > 0</math> such that for every <math>x</math> in the domain of <math>f</math>, &nbsp; <math> \displaystyle \ |x-x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \varepsilon.</math> In simple English,  <math>\displaystyle f(x)</math> is continuous at a point <math>\displaystyle x = x_0</math> if for each <math>x</math> close enough to <math>x_0</math>, the function value <math>f(x)</math> is very close to <math>f(x_0)</math>, where the "close enough" restriction typically depends on the desired closeness of <math>f(x_0)</math>  to <math>f(x).</math>
Using this definition, he proved the [[intermediate value theorem|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.

=== Calculus of variations ===

Weierstrass also made advances 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 that paved the way for the modern study of the calculus of variations. Among several 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 extremum and allows one to find a minimizing curve for a given integral.

=== Other analytical theorems ===
<!--The items listed here are important and ought to be described in the article-->
{{see also|List of things named after Karl Weierstrass}}
* [[Bolzano–Weierstrass theorem]]
* [[Stone–Weierstrass theorem]]
* [[Casorati–Weierstrass theorem]]
* [[Weierstrass elliptic function]]
* [[Weierstrass function]]
* [[Weierstrass M-test]]
* [[Weierstrass preparation theorem]]
* [[Lindemann–Weierstrass theorem]]
* [[Weierstrass factorization theorem]]
* [[Weierstrass–Enneper parameterization]]