Editing Intermediate value theorem (section)

==Generalizations==

=== Multi-dimensional spaces ===
The [[Poincaré-Miranda theorem]] is a generalization of the Intermediate value theorem from a (one-dimensional) interval to a (two-dimensional) rectangle, or more generally, to an ''n''-dimensional [[N-cube|cube]].

Vrahatis<ref>{{Cite journal |last=Vrahatis |first=Michael N. |date=2016-04-01 |title=Generalization of the Bolzano theorem for simplices |url=https://www.sciencedirect.com/science/article/pii/S0166864115005994 |journal=Topology and Its Applications |language=en |volume=202 |pages=40–46 |doi=10.1016/j.topol.2015.12.066 |issn=0166-8641}}</ref> presents a similar generalization to triangles, or more generally, ''n''-dimensional [[Simplex|simplices]]. Let ''D<sup>n</sup>'' be an ''n''-dimensional simplex with ''n''+1 vertices denoted by ''v''<sub>0</sub>,...,''v<sub>n</sub>''. Let ''F''=(''f''<sub>1</sub>,...,''f<sub>n</sub>'') be a continuous function from ''D<sup>n</sup>'' to ''R<sup>n</sup>'', that never equals 0 on the boundary of ''D<sup>n</sup>''. Suppose ''F'' satisfies the following conditions:

* For all ''i'' in 1,...,''n'', the sign of ''f<sub>i</sub>''(''v<sub>i</sub>'') is opposite to the sign of ''f<sub>i</sub>''(''x'') for all points ''x'' on the face opposite to ''v<sub>i</sub>'';
* The sign-vector of ''f''<sub>1</sub>,...,''f<sub>n</sub>'' on ''v''<sub>0</sub> is not equal to the sign-vector of ''f''<sub>1</sub>,...,''f<sub>n</sub>''  on all points on the face opposite to ''v<sub>0</sub>''.

Then there is a point ''z'' in the [[Interior (topology)|interior]] of ''D<sup>n</sup>'' on which ''F''(''z'')=(0,...,0).

It is possible to normalize the ''f<sub>i</sub>'' such that ''f<sub>i</sub>''(''v<sub>i</sub>'')>0 for all ''i''; then the conditions become simpler:

*For all ''i'' in 1,...,''n'', ''f<sub>i</sub>''(''v<sub>i</sub>'')>0, and ''f<sub>i</sub>''(''x'')<0 for all points ''x'' on the face opposite to ''v<sub>i</sub>''. In particular, ''f<sub>i</sub>''(''v<sub>0</sub>'')<0.
*For all points ''x'' on the face opposite to ''v<sub>0</sub>'', ''f<sub>i</sub>''(''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 |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 ===
The intermediate value theorem is closely linked to the [[topology|topological]] notion of [[Connectedness (topology)|connectedness]] and follows from the basic properties of connected sets in metric spaces and connected subsets of '''R''' in particular:
* If <math>X</math> and <math>Y</math> are [[metric space]]s, <math>f \colon X \to Y</math> is a continuous map, and <math>E \subset X</math> is a [[Connected space|connected]] subset, then <math>f(E)</math> is connected. ({{EquationRef|<nowiki>*</nowiki>}}) 
* A subset <math>E \subset \R</math> is connected if and only if it satisfies the following property: <math>x,y\in E,\ x < r < y \implies r \in E</math>. ({{EquationRef|<nowiki>**</nowiki>}})

In fact, connectedness is a [[topological property]] and {{EquationNote|*|(*)}} generalizes to [[topological space]]s: ''If <math>X</math> and <math>Y</math> are topological spaces, <math>f \colon X \to Y</math> is a continuous map, and <math>X</math> is a [[connected space]], then <math>f(X)</math> is connected.''  The preservation of connectedness under continuous maps can be thought of as a generalization of the intermediate value theorem, a property of continuous, real-valued functions of a real variable, to continuous functions in general spaces.

Recall the first version of the intermediate value theorem, stated previously:

{{math theorem|name=Intermediate value theorem|note=''Version I''|math_statement=Consider a closed interval <math>I = [a,b]</math> in the real numbers <math>\R</math> and a continuous function <math>f\colon I\to\R</math>. Then, if <math> u</math> is a real number such that <math>\min(f(a),f(b))< u < \max(f(a),f(b))</math>, there exists <math>c \in (a,b)</math> such that <math>f(c) = u</math>.}}

The intermediate value theorem is an immediate consequence of these two properties of connectedness:<ref>{{Cite book| url=https://archive.org/details/1979RudinW|title=Principles of Mathematical Analysis| last=Rudin|first=Walter| publisher=McGraw-Hill|year=1976|isbn=978-0-07-054235-8|location=New York|pages=42, 93}}</ref>

{{math proof|proof= By {{EquationNote|**|(**)}}, <math>I = [a,b]</math> is a connected set.  It follows from {{EquationNote|*|(*)}} that the image, <math>f(I)</math>, is also connected.  For convenience, assume that <math>f(a) < f(b)</math>.  Then once more invoking {{EquationNote|**|(**)}}, <math>f(a) < u < f(b)</math> implies that <math>u \in f(I)</math>, or <math>f(c) = u</math> for some <math>c\in I</math>.  Since <math>u\neq f(a), f(b)</math>, <math>c\in(a,b)</math> must actually hold, and the desired conclusion follows.  The same argument applies if <math>f(b) < f(a)</math>, so we are done. [[Q.E.D.]]}}

The intermediate value theorem generalizes in a natural way: Suppose that {{mvar|X}} is a connected topological space and {{math|(''Y'', <)}} is a [[total order|totally ordered]] set equipped with the [[order topology]], and let {{math|''f'' : ''X'' → ''Y''}} be a continuous map. If {{mvar|a}} and {{mvar|b}} are two points in {{mvar|X}} and {{mvar|u}} is a point in {{mvar|Y}} lying between {{math|''f''(''a'')}} and {{math|''f''(''b'')}} with respect to {{math|<}}, then there exists {{mvar|c}} in {{mvar|X}} such that {{math|1=''f''(''c'') = ''u''}}.  The original theorem is recovered by noting that {{math|'''R'''}} is connected and that its natural [[Topological space|topology]] is the order topology.

The [[Brouwer fixed-point theorem]] is a related theorem that, in one dimension, gives a special case of the intermediate value theorem.