Editing Intermediate value theorem (section)

==History==
A form of the theorem was postulated as early as the 5th century BCE, in the work of [[Bryson of Heraclea]] on [[squaring the circle]]. Bryson argued that, as circles larger than and smaller than a given square both exist, there must exist a circle of equal area.<ref>{{cite book
 | last = Bos | first = Henk J. M.
 | contribution = The legitimation of geometrical procedures before 1590
 | doi = 10.1007/978-1-4613-0087-8_2
 | mr = 1800805
 | pages = 23–36
 | publisher = Springer | location = New York
 | series = Sources and Studies in the History of Mathematics and Physical Sciences
 | title = Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction
 | year = 2001| isbn = 978-1-4612-6521-4
 }}</ref> The theorem was first proved by [[Bernard Bolzano]] in 1817. Bolzano used the following formulation of the theorem:<ref>{{Cite journal| title=A translation of Bolzano's paper on the intermediate value theorem| first=S.B.| last=Russ| journal=Historia Mathematica| date=1980| volume=7| issue=2| pages=156–185| doi=10.1016/0315-0860(80)90036-1| doi-access=free}}</ref>

Let <math>f, \varphi</math> be continuous functions on the interval between <math>\alpha</math> and <math>\beta</math> such that <math>f(\alpha) < \varphi(\alpha)</math> and <math>f(\beta) > \varphi(\beta)</math>. Then there is an <math>x</math> between <math>\alpha</math> and <math>\beta</math> such that <math>f(x) = \varphi(x)</math>.

The equivalence between this formulation and the modern one can be shown by setting <math>\varphi</math> to the appropriate [[constant function]]. [[Augustin-Louis Cauchy]] provided the modern formulation and a proof in 1821.<ref name="grabiner">{{Cite journal| 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| doi=10.2307/2975545| issue=3| jstor=2975545}}</ref> Both were inspired by the goal of formalizing the analysis of functions and the work of [[Joseph-Louis Lagrange]]. The idea that continuous functions possess the intermediate value property has an earlier origin. [[Simon Stevin]] proved the intermediate value theorem for [[polynomial]]s (using a [[Cubic function|cubic]] as an example) by providing an algorithm for constructing the decimal expansion of the solution.  The algorithm iteratively subdivides the interval into 10 parts, producing an additional decimal digit at each step of the iteration.<ref>Karin Usadi Katz and [[Mikhail Katz|Mikhail G. Katz]] (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. [[Foundations of Science]]. {{doi|10.1007/s10699-011-9223-1}} See [https://doi.org/10.1007%2Fs10699-011-9223-1 link]</ref> Before the formal definition of continuity was given, the intermediate value property was given as part of the definition of a continuous function. Proponents include [[Louis Arbogast]], who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable.<ref>{{MacTutor Biography| id=Arbogast| title=Louis François Antoine Arbogast}}</ref>
Earlier authors held the result to be intuitively obvious and requiring no proof.  The insight of Bolzano and Cauchy was to define a general notion of continuity (in terms of [[infinitesimal]]s in Cauchy's case and using real inequalities in Bolzano's case), and to provide a proof based on such definitions.