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Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
often cited as the "father of modern
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics,
botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who specialises in this field. The term "botany" comes from the Ancient Greek wo ...

and gymnastics. He later received an honorary doctorate and became professor of mathematics in Berlin. Among many other contributions, Weierstrass formalized the definition of the continuity of a function, proved the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if ''f'' is a continuous function whose domain contains the interval 'a'', ''b'' then it takes on any given value between ''f''(''a'') and ''f''(''b'') at some point ...

and the
Bolzano–Weierstrass theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, and used the latter to study the properties of continuous functions on closed bounded intervals.

# Biography

Weierstrass was born into a
Roman Catholic Roman or Romans most often refers to: *Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Laz ...

family in Ostenfelde, which was a part of
Ennigerloh Ennigerloh is a town in the district of Warendorf, in North Rhine-Westphalia, Germany ) , image_map = , map_caption = , map_width = 250px , capital = Berlin , coordinates = , largest_city = capital , languages_type = Official la ...
, in the
Province of Westphalia The Province of Westphalia () was a Provinces of Prussia, province of the Kingdom of Prussia and the Free State of Prussia from 1815 to 1946. History Napoleon I of France, Napoleon Bonaparte founded the Kingdom of Westphalia, which was a client st ...
. Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst both of whom were catholic . His interest in mathematics began while he was a
gymnasium Gymnasium may refer to: *Gymnasium (ancient Greece), educational and sporting institution *Gymnasium (school), type of secondary school that prepares students for higher education **Gymnasium (Denmark) **Gymnasium (Germany) **Gymnasium UNT, high ...
student at the in
Paderborn Paderborn () is a city in eastern North Rhine-Westphalia, Germany, capital of the Paderborn (district), Paderborn district. The name of the city derives from the river Pader (river), Pader and "born", an old German term for the source of a river. ...

. He was sent to the
University of Bonn The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university A public university or public college is a university A university ( la, universitas, 'a whole') is ...

upon graduation to prepare for a government position. Because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study but continuing private study in mathematics. The outcome was that he left the university without a degree. He then studied mathematics at the Münster Academy (which was even then famous for mathematics) and his father was able to obtain a place for him in a teacher training school in
Münster Münster ( , ; nds, Mönster) is an independent city An independent city or independent town is a city or town that does not form part of another general-purpose local government entity (such as a province). Historical precursors In the H ...

. Later he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of
Christoph Gudermann Christoph Gudermann (25 March 1798 – 25 September 1852) was a German mathematician noted for introducing the Gudermannian function of the Gudermannian function The Gudermannian function, named after Christoph Gudermann (1798–1852), relate ...
and became interested in
elliptic function In the mathematical field of complex analysis elliptic functions are a special kind of Meromorphic function, meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. ...
s. In 1843 he taught in Deutsch Krone in
West Prussia The Province of West Prussia (german: Provinz Westpreußen; csb, Zôpadné Prësë; pl, Prusy Zachodnie) was a Provinces of Prussia, province of Prussia from 1773 to 1829 and 1878 to 1922. West Prussia was established as a province of the King ...
and since 1848 he taught at the Lyceum Hosianum in
Braunsberg Braniewo () (german: Braunsberg in Ostpreußen, la, Brunsberga, Old Prussian: ''Brus'', lt, Prūsa), is a town A town is a human settlement. Towns are generally larger than villages and smaller than city, cities, though the criter ...

. Besides mathematics he also taught physics, botany, and gymnastics. Weierstrass may have had an illegitimate child named Franz with the widow of his friend
Carl Wilhelm Borchardt Carl Wilhelm Borchardt (22 February 1817 – 27 June 1880) was a Germany, German mathematics, mathematician. Borchardt was born to a Jewish family in Berlin. His father, Moritz, was a respected merchant, and his mother was Emma Heilborn. Borc ...

. After 1850 Weierstrass suffered from a long period of illness, but was able to publish mathematical articles that brought him fame and distinction. The
University of Königsberg The University of Königsberg (german: Albertus-Universität Königsberg) was the university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) educ ...
conferred an honorary doctor's degree on him on 31 March 1854. In 1856 he took a chair at the ''Gewerbeinstitut'' in Berlin (an institute to educate technical workers which would later merge with the ''Bauakademie'' to form the
Technical University of Berlin The Technical University of Berlin (official name both in English language, English and german: Technische Universität Berlin, also known as TU Berlin and Berlin Institute of Technology) is a public university, public research university locate ...
). In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became the Humboldt Universität zu Berlin. In 1870, at the age of fifty-five, Weierstrass met 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 Pneumonia is an inflammatory Inflammatory may refer to: * Inflammation, a biological response to harmful stimuli * The word ''inflammatory'' is also used to refer literally to fire and flammability, and figuratively in relation to comments t ...

.

# Mathematical contributions

## Soundness of calculus

Weierstrass was interested in the
soundness In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

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
Bolzano Bolzano ( or ; german: Bozen (formerly ), ; bar, Bozn; lld, Balsan or ) is the capital city A capital or capital city is the municipality holding primary status in a Department (country subdivision), department, country, Constituent state, ...

had developed a reasonably rigorous definition of a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

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 Limit or Limits may refer to: Arts and media * Limit (music) In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...

and continuity of functions. The basic idea behind Delta-epsilon proofs is, arguably, first found in the works of
Cauchy Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

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 that is false in general. 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 convergenceIn the mathematical field of analysis, uniform convergence is a mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections t ...
, which was first observed by Weierstrass's advisor,
Christoph Gudermann Christoph Gudermann (25 March 1798 – 25 September 1852) was a German mathematician noted for introducing the Gudermannian function of the Gudermannian function The Gudermannian function, named after Christoph Gudermann (1798–1852), relate ...
, 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: $\displaystyle f\left(x\right)$ is continuous at $\displaystyle x = x_0$ if $\displaystyle \forall \ \varepsilon > 0\ \exists\ \delta > 0$ such that for every $x$ in the domain of $f$,   $\displaystyle \ , x-x_0, < \delta \Rightarrow , f\left(x\right) - f\left(x_0\right), < \varepsilon.$ In simple English, $\displaystyle f\left(x\right)$ is continuous at a point $\displaystyle x = x_0$ if for each $x$ close enough to $x_0$, the function value $f\left(x\right)$ is very close to $f\left(x_0\right)$, where the "close enough" restriction typically depends on the desired closeness of $f\left(x_0\right)$ to $f\left(x\right).$ Using this definition, he proved the He also proved the
Bolzano–Weierstrass theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathemati ...
. 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

*
Stone–Weierstrass theoremIn mathematical analysis, the Weierstrass approximation theorem states that every continuous function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...
*
Casorati–Weierstrass theoremIn complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularity, essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice C ...
*
Weierstrass elliptic function In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as p-functions and they are usually denoted by the symbo ...
*
Weierstrass function 300px, Plot of Weierstrass function over the interval minus;2, 2 Like other fractals, the function exhibits self-similarity">fractal.html" ;"title="minus;2, 2 Like other fractal">minus;2, 2 Like other fractals, the function exhibi ...

* Weierstrass M-test *
Weierstrass preparation theorem In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point ''P''. It states that such a function is, up to multiplication by a function not zero at ''P'', a polyn ...
*
Lindemann–Weierstrass theorem In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following. In other words the extension field has transcendence degree In abst ...
*
Weierstrass factorization theoremIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
* Weierstrass–Enneper parameterization

# Students

*
Edmund Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...

# Honours and awards

The lunar
crater Crater may refer to: Landforms *Impact crater, a depression caused by two celestial bodies impacting each other, such as a meteorite hitting a planet *Explosion crater, a hole formed in the ground produced by an explosion near or below the surface ...

Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...
and the
asteroid An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Historically, these terms have been applied to any astronomical object orbiting the Sun that did not resolve into a disc in a telescope and was not observ ...

14100 Weierstrass are named after him. Also, there is the Weierstrass Institute for Applied Analysis and Stochastics in Berlin.

# Selected works

* ''Zur Theorie der Abelschen Funktionen'' (1854) * ''Theorie der Abelschen Funktionen'' (1856) *
Abhandlungen-1
', Math. Werke. Bd. 1. Berlin, 1894 *
Abhandlungen-2
', Math. Werke. Bd. 2. Berlin, 1895 *
Abhandlungen-3
', Math. Werke. Bd. 3. Berlin, 1903 *
Vorl. ueber die Theorie der Abelschen Transcendenten
', Math. Werke. Bd. 4. Berlin, 1902 *
Vorl. ueber Variationsrechnung
', Math. Werke. Bd. 7. Leipzig, 1927

* List of things named after Karl Weierstrass