Intermediate value theorem: Difference between revisions

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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> <ref>{{cite book |doi=10.1007/978-3-030-11036-9|title=Cauchy's Calcul Infinitésimal |year=2019 |last1=Cates |first1=Dennis M. |isbn=978-3-030-11035-2 |s2cid=132587955|page=249 }}</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 anthe closed interval <math>I = [a,b]</math> of real numbers <math>\R</math> and a continuous function <math>f \colon I \to \R</math>. Then

*''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 ana closed interval (closed), and it contains <math>\bigl[\min(f(a), f(b)),\max(f(a), f(b))\bigr]</math>,.

'''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>, even if they are outside the interval between <math>f(a)</math> and <math>f(b)</math>, all points in the interval <math>\bigl[c,d\bigr]</math> are also function values, <math display="block">\bigl[c,d\bigr]\subseteq f(I).</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^2-2</math> for <math>x\in\Q</math> satisfies <math>f(0) = -20</math> and <math>f(2) = 24</math>. However, there is no rational number <math>x</math> such that <math>f(x)=02</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==

=== Proof version A===

<!-- This section is linked from [[Continuity property]] -->

The theorem may be proven as a consequence of the [[completeness (order theory)|completeness]] property of the real numbers as follows:<ref>Essentially follows {{cite book |title=Foundations of Analysis|first=Douglas A.|last=Clarke|publisher=Appleton-Century-Crofts | year=1971|page=284}}</ref>

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Note that, due to the continuity of <math>f</math> at <math>a</math>, we can keep <math>f(x)</math> within any <math>\varepsilon>0</math> of <math>f(a)</math> by keeping <math>x</math> sufficiently close to <math>a</math>. Since <math>f(a)<u</math> is a strict inequality, consider the implication when <math>\varepsilon</math> is the distance between <math>u</math> and <math>f(a)</math>. No <math>x</math> sufficiently close to <math>a</math> can then make <math>f(x)</math> greater than or equal to <math>u</math>, which means there are values greater than <math>a</math> in <math>S</math>. A more detailed proof goes like this:

Choose <math>\varepsilon=u-f(a)>0</math>. Then <math>\exists \delta>0</math> such that <math>\forall x \in [a,b]</math>, <math display="block">|x-a|<\delta \implies |f(x)-f(a)|<u-f(a) \implies f(x)<u.</math>Consider the interval <math>[a,\min(a+\delta,b))=I_1</math>. Notice that <math>I_1 \subseteq [a,b]</math> and every <math>x \in I_1</math> satisfies the condition <math>|x-a|<\delta</math>. Therefore for every <math>x \in I_1</math> we have <math>f(x)<u</math>. Hence <math>c</math> cannot be <math>a</math>.

Consider the interval <math>[a,\min(a+\delta,b))=I_1</math>. Notice that <math>I_1 \subseteq [a,b]</math> and every <math>x \in I_1</math> satisfies the condition <math>|x-a|<\delta</math>. Therefore for every <math>x \in I_1</math> we have <math>f(x)<u</math>. Hence <math>c</math> cannot be <math>a</math>.

Likewise, due to the continuity of <math>f</math> at <math>b</math>, we can keep <math>f(x)</math> within any <math>\varepsilon > 0</math> of <math>f(b)</math> by keeping <math>x</math> sufficiently close to <math>b</math>. Since <math>u<f(b)</math> is a strict inequality, consider the similar implication when <math>\varepsilon</math> is the distance between <math>u</math> and <math>f(b)</math>. Every <math>x</math> sufficiently close to <math>b</math> must then make <math>f(x)</math> greater than <math>u</math>, which means there are values smaller than <math>b</math> that are upper bounds of <math>S</math>. A more detailed proof goes like this:

Choose <math>\varepsilon=f(b)-u>0</math>. Then <math>\exists \delta>0</math> such that <math>\forall x \in [a,b]</math>, <math display="block">|x-b|<\delta \implies |f(x)-f(b)|<f(b)-u \implies f(x)>u.</math>Consider the interval <math>(\max(a,b-\delta),b]=I_2</math>. Notice that <math>I_2 \subseteq [a,b]</math> and every <math>x \in I_2</math> satisfies the condition <math>|x-b|<\delta</math>. Therefore for every <math>x \in I_2</math> we have <math>f(x)>u</math>. Hence <math>c</math> cannot be <math>b</math>.

Consider the interval <math>(\max(a,b-\delta),b]=I_2</math>. Notice that <math>I_2 \subseteq [a,b]</math> and every <math>x \in I_2</math> satisfies the condition <math>|x-b|<\delta</math>. Therefore for every <math>x \in I_2</math> we have <math>f(x)>u</math>. Hence <math>c</math> cannot be <math>b</math>.

With <math>c \neq a</math> and <math>c \neq b</math>, it must be the case <math>c \in (a,b)</math>. Now we claim that <math>f(c)=u</math>.

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<math display="block">u-\varepsilon<f(c)< u+\varepsilon</math>

are valid for all <math>\varepsilon > 0</math>, from which we deduce <math>f(c) = u</math> as the only possible value, as stated.

===Proof version B===

We will only prove the case of <math>f(a)<u<f(b)</math>, as the <math>f(a)>u>f(b)</math> case is similar.<ref>Slightly modified version of {{cite book |title=Understanding Analysis|first=Stephen|last=Abbot|publisher=Springer | year=2015|page=123}}</ref>

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

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

The equivalence between this formulation and the modern one can be shown by setting <math>\phivarphi</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,&thinsp;∞) → [−1,&thinsp;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 the [[Conway's base 13 function]].

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).

Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions;<ref>{{Cite book |last=Smorynski |first=Craig |url=https://books.google.com/books?id=lnuhDgAAQBAJ&pg=PA51&q=Historically%2C+this+intermediate+value+property+has+been+suggested+as+a+definition+for+continuity+of+real-valued+functions&pg=PA51 |title=MVT: A Most Valuable Theorem |date=2017-04-07 |publisher=Springer |isbn=9783319529561 |language=en}}</ref> this definition was not adopted.

==Generalizations==

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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 itsIts 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 ''Dⁿ'' be an ''n''-dimensional simplex with ''n''+1 vertices denoted by ''v''₀,...,''v_n''. Let ''F''=(''f''₁,...,''f_n'') be a continuous function from ''Dⁿ'' to ''Rⁿ'', that never equals 0 on the boundary of ''Dⁿ''. Suppose ''F'' satisfies the following conditions:

* For all ''i'' in 1,...,''n'', the sign of ''f_i''(''v_i'') is opposite to the sign of ''f_i''(''x'') for all points ''x'' on the face opposite to ''v_i'';

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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 itsIts 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|<mathnowiki>(*)</mathnowiki>}})

* 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|<mathnowiki>(**)</mathnowiki>}})

In fact, connectedness is a [[topological property]] and <math>{{EquationNote|*|(*)</math>}} 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 realcontinuous, real-valued [[Function of a real variable|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 <math>{{EquationNote|**|(**)</math>}}, <math>I = [a,b]</math> is a connected set. It follows from <math>{{EquationNote|*|(*)</math>}} that the image, <math>f(I)</math>, is also connected. For convenience, assume that <math>f(a) < f(b)</math>. Then once more invoking <math>{{EquationNote|**|(**)</math>}}, <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.

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==In constructive mathematics==

In [[constructive mathematics]], the intermediate value theorem is not true. Instead, the weakened conclusion one hasmust totake weakenstates that the conclusion:value may only be found in some range which may be arbitrarily small.

* 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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In general, for any continuous function whose ___domain is some closed convex {{nowrap|<math>n</math>-dimensional}} shape and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same.

The theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily met constraints).<ref>[[Keith Devlin]] (2007) [https://web.archive.org/web/20140228044921/http://www.maa.org/external_archive/devlin/devlin_02_07.html How to stabilize a wobbly table]</ref>

==See also==

* {{annotated link|[[Mean value theorem}}]]

* {{annotated link|[[Non-atomic measure}}]]

* {{annotated link|[[Hairy ball theorem}}]]

* {{annotated link|[[Sperner's lemma}}]]

==References==

{{Reflist}}

== Further reading ==

* https://mathoverflow.net/questions/253059/approximate-intermediate-value-theorem-in-pure-constructive-mathematics

==External links==

{{ProofWiki|id=Intermediate_Value_Theorem|title=Intermediate value Theorem}}

* [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]].