In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval and is a number such that , then there exists some ' between and such that . That is, the image of a continuous function over an interval is itself an interval that contains .
For example, suppose that ' , then the graph of must pass through the horizontal line while moves from to . Over the interval, the set of function values has no gap, and the graph can be drawn without lifting a pencil from the paper.
The corollary Bolzano's theorem states that if a continuous function has values of opposite sign inside an interval, then it has a root in that interval. The theorem depends on, and is equivalent to, the completeness of the real numbers, although Weierstrass Nullstellensatz is a version of the intermediate value theorem for polynomials over a real closed field.
A similar result to the intermediate value theorem is the BorsukâÂÂUlam theorem, which underpins why rotating a wobbly table will always bring it to stability. Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property, even though they need not be continuous.
This captures an intuitive property of continuous functions over the real numbers: given ' continuous on with the known values and , then the graph of must pass through the horizontal line while moves from to . 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.
The intermediate value theorem states the following:
Consider the closed interval of real numbers and a continuous function . Then
Version I is naturally contained in Version II.
The theorem depends on, and is equivalent to, the completeness of the real numbers. The intermediate value theorem does not apply to the rational numbers Q because gaps exist between rational numbers; irrational numbers fill those gaps. For example, the function for satisfies and . However, there is no rational number such that , because 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.
The theorem may be proven as a consequence of the completeness property of the real numbers as follows:
We shall prove the first case, . The second case is similar.
A more detailed proof goes like this:
An even more detailed proof goes like this:
We will only prove the case of , as the case is similar.
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 footing.
A form of the theorem was postulated as early as the 5th century BCE, in the work of Bryson of Heraclea on squaring the circle. Bryson argued that, as circles larger than and smaller than a given square both exist, there must exist a circle of equal area. The theorem was first proved by Bernard Bolzano in 1817. Bolzano used the following formulation of the theorem:<blockquote>Let be continuous functions on the interval between and such that and . Then there is an between and such that .</blockquote>The equivalence between this formulation and the modern one can be shown by setting to the appropriate constant function. Augustin-Louis Cauchy 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. The idea that continuous functions possess the intermediate value property has an earlier origin. Simon Stevin proved the intermediate value theorem for polynomials (using a 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. 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 infinitesimals in Cauchy's case and using real inequalities in Bolzano's case), and to provide a proof based on such definitions.
A Darboux function is a real-valued function that has the "intermediate value property," i.e., that satisfies the conclusion of the intermediate value theorem: for any two values and in the domain of , and any between and , there is some between and with . 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 defined by for and . This function is not continuous at because the limit of as tends to 0 does not exist; yet the function has the intermediate value property. Another, more complicated example is given by Conway's base 13 function.
In fact, Darboux's theorem states that all functions that result from the 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.
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 cube.
Vrahatis presents a similar generalization to triangles, or more generally, n-dimensional 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:
Then there is a point z in the 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:
The theorem can be proved based on the KnasterâÂÂKuratowskiâÂÂMazurkiewicz lemma. In can be used for approximations of fixed points and zeros.
The intermediate value theorem is closely linked to the topological notion of connectedness and follows from the basic properties of connected sets in metric spaces and connected subsets of R in particular:
In fact, connectedness is a topological property and generalizes to topological spaces: If and are topological spaces, is a continuous map, and is a connected space, then 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:
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 is a connected topological space and is a totally ordered set equipped with the order topology, and let be a continuous map. If and are two points in and is a point in lying between and with respect to , then there exists in such that . The original theorem is recovered by noting that is connected and that its natural 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.
In constructive mathematics, the intermediate value theorem is not true. Instead, the weakened conclusion one must take states that the value may only be found in some range which may be arbitrarily small.
A similar result is the BorsukâÂÂUlam theorem, which says that a continuous map from the -sphere to Euclidean -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).