Editing Mathematical analysis (section)

===Modern===
====Foundations====
The modern foundations of mathematical analysis were established in 17th century Europe.<ref name=analysis/> This began when [[Fermat]] and [[Descartes]] developed [[analytic geometry]], which is the precursor to modern calculus. Fermat's method of [[adequality]] allowed him to determine the maxima and minima of functions and the tangents of curves.<ref name=Pellegrino>{{cite web | last = Pellegrino | first = Dana | title = Pierre de Fermat | url = http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html | access-date = 2008-02-24 | archive-date = 2008-10-12 | archive-url = https://web.archive.org/web/20081012024028/http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html | url-status = live }}</ref> Descartes's publication of ''[[La Géométrie]]'' in 1637, which introduced the [[Cartesian coordinate system]], is considered to be the establishment of mathematical analysis. It would be a few decades later that [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] independently developed [[infinitesimal calculus]], which grew, with the stimulus of applied work that continued through the 18th&nbsp;century, into analysis topics such as the [[calculus of variations]], [[Ordinary differential equation|ordinary]] and [[partial differential equation]]s, [[Fourier analysis]], and [[generating function]]s. During this period, calculus techniques were applied to approximate [[discrete mathematics|discrete problems]] by continuous ones.

====Modernization====
In the 18th century, [[Leonhard Euler|Euler]] introduced the notion of a [[function (mathematics)|mathematical function]].<ref name="function">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All| url = https://archive.org/details/eulermasterofusa0000dunh| url-access = registration| date = 1999| publisher =The Mathematical Association of America | page= [https://archive.org/details/eulermasterofusa0000dunh/page/17 17]}}</ref> Real analysis began to emerge as an independent subject when [[Bernard Bolzano]] introduced the modern definition of continuity in 1816,<ref>*{{cite book |first=Roger |last=Cooke |author-link=Roger Cooke (mathematician) |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |date=1997 |isbn=978-0471180821 |page=[https://archive.org/details/historyofmathema0000cook/page/379 379] |chapter=Beyond the Calculus |quote=Real analysis began its growth as an independent subject with the introduction of the modern definition of continuity in 1816 by the Czech mathematician Bernard Bolzano (1781–1848) |chapter-url=https://archive.org/details/historyofmathema0000cook/page/379 }}</ref> but Bolzano's work did not become widely known until the 1870s. In 1821, [[Augustin Louis Cauchy|Cauchy]] began to put calculus on a firm logical foundation by rejecting the principle of the [[generality of algebra]] widely used in earlier work, particularly by Euler.  Instead, Cauchy formulated calculus in terms of geometric ideas and [[infinitesimal]]s.  Thus, his definition of continuity required an infinitesimal change in ''x'' to correspond to an infinitesimal change in ''y''.  He also introduced the concept of the [[Cauchy sequence]], and started the formal theory of [[complex analysis]]. [[Siméon Denis Poisson|Poisson]], [[Joseph Liouville|Liouville]], [[Joseph Fourier|Fourier]] and others studied partial differential equations and [[harmonic analysis]].  The contributions of these mathematicians and others, such as [[Karl Weierstrass|Weierstrass]], developed the [[(ε, δ)-definition of limit]] approach, thus founding the modern field of mathematical analysis. Around the same time, [[Bernhard Riemann|Riemann]] introduced his theory of [[integral|integration]], and made significant advances in complex analysis.

Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a [[Continuum (set theory)|continuum]] of [[real number]]s without proof. [[Richard Dedekind|Dedekind]] then constructed the real numbers by [[Dedekind cut]]s, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a [[complete metric space|complete]] set: the continuum of real numbers, which had already been developed by [[Simon Stevin]] in terms of [[decimal expansion]]s. Around that time, the attempts to refine the [[theorem]]s of [[Riemann integral|Riemann integration]] led to the study of the "size" of the set of [[Classification of discontinuities|discontinuities]] of real functions.

Also, various [[pathological (mathematics)|pathological objects]], (such as [[nowhere continuous function]]s, continuous but [[Weierstrass function|nowhere differentiable functions]], and [[space-filling curve]]s), commonly known as "monsters", began to be investigated. In this context, [[Camille Jordan|Jordan]] developed his theory of [[Jordan measure|measure]], [[Georg Cantor|Cantor]] developed what is now called [[naive set theory]], and [[René-Louis Baire|Baire]] proved the [[Baire category theorem]]. In the early 20th century, calculus was formalized using an axiomatic [[set theory]]. [[Henri Lebesgue|Lebesgue]] greatly improved measure theory, and introduced his own theory of integration, now known as [[Lebesgue integration]], which proved to be a big improvement over Riemann's. [[David Hilbert|Hilbert]] introduced [[Hilbert space]]s to solve [[integral equation]]s. The idea of [[normed vector space]] was in the air, and in the 1920s [[Stefan Banach|Banach]] created [[functional analysis]].