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This has two important [[corollary|corollaries]]:
# If a continuous function has values of opposite sign inside an interval, then it has a [[Zero of a function|root]] in that interval ('''Bolzano's theorem''').<ref>{{MathWorld |title=Bolzano's Theorem |urlname=BolzanosTheorem}}</ref>
# The [[image (mathematics)|image]] of a continuous function over an interval is itself an interval.
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The intermediate value theorem states the following:
Consider
*''Version I.'' if <math>u</math> is a number between <math>f(a)</math> and <math>f(b)</math>, that is, <math display="block">\min(f(a),f(b))<u<\max(f(a),f(b)),</math> then there is a <math>c\in (a,b)</math> such that <math>f(c)=u</math>.
*''Version II.'' the [[Image of a function|image set]] <math>f(I)</math> is also
'''Remark:''' ''Version II'' states that the [[Set (mathematics)|set]] of function values has no gap. For any two function values <math>c,d \in f(I)</math> with <math>c < d</math>
A subset of the real numbers with no internal gap is an interval. ''Version I'' is naturally contained in ''Version II''.
==Relation to completeness==
The theorem depends on, and is equivalent to, the [[completeness of the real numbers]]. The intermediate value theorem does not apply to the [[rational number]]s '''Q''' because gaps exist between rational numbers; [[irrational numbers]] fill those gaps. For example, the function <math>f(x) = x^2</math> for <math>x\in\Q</math> satisfies <math>f(0) = 0</math> and <math>f(2) = 4</math>. However, there is no rational number <math>x</math> such that <math>f(x)=2</math>, because <math>\sqrt 2</math> is an irrational number.
Despite the above, there is a version of the intermediate value theorem for polynomials over a [[real closed field]]; see the [[Weierstrass Nullstellensatz]].
==Proof==
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Define <math>g(x)=f(x)-u</math> which is equivalent to <math>f(x)=g(x)+u</math> and lets us rewrite <math>f(a)<u<f(b)</math> as <math>g(a)<0<g(b)</math>, and we have to prove, that <math>g(c)=0</math> for some <math>c\in[a,b]</math>, which is more intuitive. We further define the set <math>S=\{x\in[a,b]:g(x)\leq 0\}</math>. Because <math>g(a)<0</math> we know, that <math>a\in S</math> so, that <math>S</math> is not empty. Moreover, as <math>S\subseteq[a,b]</math>, we know that <math>S</math> is bounded and non-empty, so by Completeness, the [[supremum]] <math>c=\sup(S)</math> exists.
There are 3 cases for the value of <math>g(c)</math>, those being <math>g(c)<0,g(c)>0</math> and <math>g(c)=0</math>. For contradiction, let us assume, that <math>g(c)<0</math>. Then, by the definition of continuity, for <math>\epsilon=0-g(c)</math>, there exists a <math>\delta>0</math> such that <math>x\in(c-\delta,c+\delta)</math> implies, that <math>|g(x)-g(c)|<-g(c)</math>, which is equivalent to <math>g(x)<0</math>. If we just chose <math>x=c+\frac{\delta}{N}</math>, where <math>N>\frac{\delta}{b-c}+1</math>, then as <math>1 < N</math>, <math>x<c+\delta</math>, from which we get <math>g(x)<0</math> and <math>c<x<b</math>, so <math>x\in S</math>. It follows that <math>x</math> is an upper bound for <math>S</math>. However, <math>x>c</math>, contradicting the '''upper bound''' property of the ''least upper bound'' <math>c</math>, so <math>g(c)\geq 0</math>. Assume then, that <math>g(c)>0</math>. We similarly chose <math>\epsilon=g(c)-0</math> and know, that there exists a <math>\delta>0</math> such that <math>x\in(c-\delta,c+\delta)</math> implies <math>|g(x)-g(c)|<g(c)</math>. We can rewrite this as <math>-g(c)<g(x)-g(c)<g(c)</math> which implies, that <math>g(x)>0</math>. If we now chose <math>x=c-\frac{\delta}{2}</math>, then <math>g(x)>0</math> and <math>a<x<c</math>. It follows that <math>x</math> is an upper bound for <math>S</math>. However, <math>x<c</math>, which contradict the '''least''' property of the ''least upper bound'' <math>c</math>, which means, that <math>g(c)>0</math> is impossible. If we combine both results, we get that <math>g(c)=0</math> or <math>f(c)=u</math> is the only remaining possibility.
'''Remark:''' The intermediate value theorem can also be proved using the methods of [[non-standard analysis]], which places "intuitive" arguments involving infinitesimals on a rigorous{{Clarify|reason=The placement and phrasing of this remark may suggest that the classical proof is somehow "intuitive" and not rigorous, which is not the case.|date=January 2023}} footing.<ref>{{cite arXiv |last=Sanders|first=Sam | eprint=1704.00281 | title=Nonstandard Analysis and Constructivism!|date=2017|class=math.LO}}</ref>
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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
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.
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A [[Darboux function]] is a real-valued function {{mvar|f}} that has the "intermediate value property," i.e., that satisfies the conclusion of the intermediate value theorem: for any two values {{mvar|a}} and {{mvar|b}} in the ___domain of {{mvar|f}}, and any {{mvar|y}} between {{math|''f''(''a'')}} and {{math|''f''(''b'')}}, there is some {{mvar|c}} between {{mvar|a}} and {{mvar|b}} with {{math|1=''f''(''c'') = ''y''}}. The intermediate value theorem says that every continuous function is a Darboux function. However, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false.
As an example, take the function {{math|''f'' : [0, ∞) → [−1, 1]}} defined by {{math|1=''f''(''x'') = sin(1/''x'')}} for {{math|''x'' > 0}} and {{math|1=''f''(0) = 0}}. This function is not continuous at {{math|1=''x'' = 0}} because the [[limit of a function|limit]] of {{math|1=''f''(''x'')}} as {{mvar|x}} tends to 0 does not exist; yet the function has the intermediate value property. Another, more complicated example is given by
In fact, [[Darboux's theorem (analysis)|Darboux's theorem]] states that all functions that result from the [[derivative|differentiation]] of some other function on some interval have the [[intermediate value property]] (even though they need not be continuous).
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*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
=== 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
Recall the first version of the intermediate value theorem, stated previously:
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==In constructive mathematics==
In [[constructive mathematics]], the intermediate value theorem is not true. Instead, the weakened conclusion one
* Let <math>a</math> and <math>b</math> be real numbers and <math>f:[a,b] \to R</math> be a pointwise continuous function from the [[closed interval]] <math>[a,b]</math> to the real line, and suppose that <math>f(a) < 0</math> and <math>0 < f(b)</math>. Then for every positive number <math>\varepsilon > 0</math> there exists a point <math>x</math> in the unit interval such that <math>\vert f(x) \vert < \varepsilon</math>.<ref>{{cite journal|title=Interpolating Between Choices for the Approximate Intermediate Value Theorem | author=Matthew Frank|journal=Logical Methods in Computer Science|volume=16|issue=3|date=July 14, 2020| doi=10.23638/LMCS-16(3:5)2020|arxiv=1701.02227}}</ref>
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==See also==
*
*
*
*
==References==
{{Reflist}}
== Further reading ==
* https://mathoverflow.net/questions/253059/approximate-intermediate-value-theorem-in-pure-constructive-mathematics
==External links==
* [http://www.cut-the-knot.org/Generalization/ivt.shtml Intermediate value Theorem - Bolzano Theorem] at [[cut-the-knot]]
* [http://demonstrations.wolfram.com/BolzanosTheorem/ Bolzano's Theorem] by Julio Cesar de la Yncera, [[Wolfram Demonstrations Project]].
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