TheInfoList In
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, the intermediate value theorem states that if ''f'' is a continuous
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
contains the interval , then it takes on any given value between ''f''(''a'') and ''f''(''b'') at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a
root In vascular plant Vascular plants (from Latin ''vasculum'': duct), also known as Tracheophyta (the tracheophytes , from Greek τραχεῖα ἀρτηρία ''trācheia artēria'' 'windpipe' + φυτά ''phutá'' 'plants'), form a large grou ...
in that interval (Bolzano's theorem). # The
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of a continuous function over an interval is itself an interval.

# Motivation

This captures an intuitive property of continuous functions over the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s: given ''f'' continuous on , 2with the known values ''f''(1) = 3 and ''f''(2) = 5, then the graph of ''y'' = ''f''(''x'') must pass through the horizontal line ''y'' = 4 while ''x'' moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.

# Theorem

The intermediate value theorem states the following: Consider an interval

# Relation to completeness

The theorem depends on, and is equivalent to, the
completeness of the real numbers Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line In mathematics, the real line, or real number line is the line (geometry), line whose Point (geometr ...
. The intermediate value theorem does not apply to the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s Q because gaps exist between rational numbers;
irrational numbers In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
fill those gaps. For example, the function $f\left(x\right)=x^2-2$ for $x\in\Q$ satisfies $f\left(0\right)=-2$ and $f\left(2\right)=2$. However, there is no rational number $x$ such that $f\left(x\right)=0$, because $\sqrt2$ is an irrational number.

# Proof

The theorem may be proven as a consequence of the completeness property of the real numbers as follows: We shall prove the first case,
supremum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... $c=\sup S$ exists. That is, $c$ is the smallest number that is greater than or equal to every member of $S$. We claim that $f\left(c\right)=u$. Fix some $\varepsilon > 0$. Since $f$ is continuous, there is a $\delta>0$ such that $, f\left(x\right) - f\left(c\right), < \varepsilon$ whenever $, x-c, < \delta$. This means that :
non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard ...
, which places "intuitive" arguments involving infinitesimals on a rigorous footing.

# History

The theorem was first proved by
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian A Bohemian () is a resident of Bohemia Bohemia ( ; cs, Čechy ; ; hsb, Čěska; szl, Czechy) is the westernmost a ... in 1817. Bolzano used the following formulation of the theorem: Let $f, \phi$ be continuous functions on the interval between $\alpha$ and $\beta$ such that $f\left(\alpha\right) < \phi\left(\alpha\right)$ and $f\left(\beta\right) > \phi\left(\beta\right)$. Then there is an $x$ between $\alpha$ and $\beta$ such that $f\left(x\right) = \phi\left(x\right)$. The equivalence between this formulation and the modern one can be shown by setting $\phi$ to the appropriate constant function.
Augustin-Louis Cauchy Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ... provided the modern formulation and a proof in 1821. Both were inspired by the goal of formalizing the analysis of functions and the work of
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaSimon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
proved the intermediate value theorem for
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... s (using a
cubic Cubic may refer to: Science and mathematics * Cube (algebra) In arithmetic and algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathema ... 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. 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. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s in Cauchy's case and using real inequalities in Bolzano's case), and to provide a proof based on such definitions.

# Generalizations

The intermediate value theorem is closely linked to the
topological s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ... notion of
connectedness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and follows from the basic properties of connected sets in metric spaces and connected subsets of R in particular: * If $X$ and $Y$ are
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s, $f \colon X \to Y$ is a continuous map, and $E \subset X$ is a connected subset, then $f\left(E\right)$ is connected. (*) * A subset $E \subset \R$ is connected if and only if it satisfies the following property: $x,y\in E,\ x < r < y \implies r \in E$. (**) In fact, connectedness is a
topological property In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
and (*) generalizes to
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s: ''If $X$ and $Y$ are topological spaces, $f \colon X \to Y$ is a continuous map, and $X$ is a
connected space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
, then $f\left(X\right)$ is connected.'' The preservation of connectedness under continuous maps can be thought of as a generalization of the intermediate value theorem, a property of real valued functions of a real variable, to continuous functions in general spaces. Recall the first version of the intermediate value theorem, stated previously: The intermediate value theorem is an immediate consequence of these two properties of connectedness: The intermediate value theorem generalizes in a natural way: Suppose that ''X'' is a connected topological space and (''Y'', <) is a
totally ordered In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
set equipped with the
order topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and let ''f'' : ''X'' → ''Y'' be a continuous map. If ''a'' and ''b'' are two points in ''X'' and ''u'' is a point in ''Y'' lying between ''f''(''a'') and ''f''(''b'') with respect to <, then there exists ''c'' in ''X'' such that ''f''(''c'') = ''u''. The original theorem is recovered by noting that R is connected and that its natural
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
is the order topology. The
Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compactness, compact convex set to itself there is a po ...
is a related theorem that, in one dimension, gives a special case of the intermediate value theorem.

# Converse is false

A Darboux function is a real-valued function ''f'' that has the "intermediate value property," i.e., that satisfies the conclusion of the intermediate value theorem: for any two values ''a'' and ''b'' in the domain of ''f'', and any ''y'' between ''f''(''a'') and ''f''(''b''), there is some ''c'' between ''a'' and ''b'' with ''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 ''f'' : , ∞)_→_[−1, 1defined_by_''f''(''x'') = sin(1/''x'')_for_''x'' > 0_and_''f''(0) = 0._This_function_is_not_continuous_at_''x'' = 0_because_the_limit_of_a_function.html" "title="��1, 1.html" ;"title=", ∞) → [−1, 1">, ∞) → [−1, 1defined by ''f''(''x'') = sin(1/''x'') for ''x'' > 0 and ''f''(0) = 0. This function is not continuous at ''x'' = 0 because the limit of a function">limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
of ''f''(''x'') as ''x'' tends to 0 does not exist; yet the function has the intermediate value property. Another, more complicated example is given by the Conway 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; this definition was not adopted.

# Practical applications

A similar result is the
Borsuk–Ulam theorem In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere, ''n''-sphere into Euclidean space, Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called ...
, which says that a continuous map from the $n$-sphere to Euclidean $n$-space will always map some pair of antipodal points to the same place. In general, for any continuous function whose domain is some closed convex 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).
Keith Devlin Keith J. Devlin (born 16 March 1947) is a British mathematician and popular science Popular science (also called pop-science or popsci) is an interpretation of science intended for a general audience. While science journalism focuses on recent ...
(2007
How to stabilize a wobbly table
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* * *

# References

Intermediate value Theorem - Bolzano Theorem
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electron ...

Bolzano's Theorem
by Julio Cesar de la Yncera,
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hoste ...
. * * {{cite web , url=https://math.stackexchange.com/q/95867 , title=Two-dimensional version of the Intermediate Value Theorem , first=Jim , last=Belk , work=
Stack Exchange Stack Exchange is a network of question-and-answer (Q&A) websites A website (also written as web site) is a collection of web pages and related content that is identified by a common domain name and published on at least one web server. ...
, date=January 2, 2012 *
Mizar system The Mizar system consists of a formal language In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
proof: http://mizar.org/version/current/html/topreal5.html#T4 Continuous mappings Articles containing proofs Theorems in calculus Theorems in real analysis