Intermediate value theorem: Difference between revisions

Revision as of 17:20, 4 December 2024

Revision as of 18:33, 12 December 2024

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| 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

| year = 2001}}</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>.

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*For all ''i'' in 1,...,''n'', ''f_i''(''v_i'')>0, and ''f_i''(''x'')<0 for all points ''x'' on the face opposite to ''v_i''. In particular, ''f_i''(''v₀'')<0.

*For all points ''x'' on the face opposite to ''v₀'', ''f_i''(''x'')>0 for at least one ''i'' in 1,...,''n.''

The theorem can be proved based on the [[Knaster–Kuratowski–Mazurkiewicz lemma]]. In can be used for approximations of fixed points and zeros.<ref>{{Cite journal |last=Vrahatis |first=Michael N. |date=2020-04-15 |title=Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros |url=https://www.sciencedirect.com/science/article/pii/S0166864119304420 |journal=Topology and Its Applications |language=en |volume=275 |pages=107036 |doi=10.1016/j.topol.2019.107036 |issn=0166-8641|doi-access=free }}</ref>

=== General metric and topological spaces ===