Continuous function on an interval takes on every value between its values at the ends
In mathematical analysis, the intermediate value theorem states that if formula_1 is a continuous function whose domain contains the interval ["a", "b"], then it takes on any given value between formula_2 and formula_3 at some point within the interval.
This has two important corollaries:
Motivation.
This captures an intuitive property of continuous functions over the real numbers: given "formula_1" continuous on formula_5 with the known values formula_6 and formula_7, then the graph of formula_8 must pass through the horizontal line formula_9 while formula_10 moves from formula_11 to formula_12. 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 formula_13 of real numbers formula_14 and a continuous function formula_15. Then
Remark: "Version II" states that the set of function values has no gap. For any two function values formula_23 with formula_24 all points in the interval formula_25 are also function values, formula_26
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 numbers Q because gaps exist between rational numbers; irrational numbers fill those gaps. For example, the function formula_27 for formula_28 satisfies formula_29 and formula_30. However, there is no rational number formula_10 such that formula_32, because formula_33 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.
The theorem may be proven as a consequence of the completeness property of the real numbers as follows:
We shall prove the first case, formula_34. The second case is similar.
Let formula_35 be the set of all formula_36 such that ;math ;f(x). Then formula_35 is non-empty since formula_38 is an element of formula_35. Since formula_35 is non-empty and bounded above by formula_41, by completeness, the supremum formula_42 exists. That is, formula_43 is the smallest number that is greater than or equal to every member of formula_35.
Note that, due to the continuity of formula_1 at formula_38, we can keep formula_47 within any formula_48 of formula_2 by keeping formula_10 sufficiently close to formula_38. Since formula_52 is the distance between formula_16 and formula_2. No formula_10 sufficiently close to formula_38 can then make formula_47 greater than or equal to formula_16, which means there are values greater than formula_38 in formula_35. A more detailed proof goes like this:
Choose formula_61. Then formula_62 such that formula_63, formula_64. Notice that formula_65 and every formula_66 satisfies the condition formula_67. Therefore for every formula_66 we have ;math ;f(x). Hence formula_43 cannot be formula_38.
Likewise, due to the continuity of formula_1 at formula_41, we can keep formula_47 within any formula_74 of formula_3 by keeping formula_10 sufficiently close to formula_41. Since formula_78 is a strict inequality, consider the similar implication when formula_79 is the distance between formula_16 and formula_3. Every formula_10 sufficiently close to formula_41 must then make formula_47 greater than formula_16, which means there are values smaller than formula_41 that are upper bounds of formula_35. A more detailed proof goes like this:
Choose formula_88. Then formula_62 such that formula_63, formula_91Consider the interval formula_92. Notice that formula_93 and every formula_94 satisfies the condition formula_95. Therefore for every formula_94 we have formula_97. Hence formula_43 cannot be formula_41.
With formula_100 and formula_101, it must be the case formula_102. Now we claim that formula_20.
Fix some formula_74. Since formula_1 is continuous at formula_43, formula_107 such that formula_63, formula_109.
Since formula_102 and formula_111 is open, formula_112 such that formula_113. Set formula_114. Then we have
formula_115
for all formula_116. By the properties of the supremum, there exists some formula_117 that is contained in formula_35, and so
formula_119, we know that formula_120 because formula_43 is the supremum of formula_35. This means that
formula_123
Both inequalities
formula_124
are valid for all formula_74, from which we deduce formula_126 as the only possible value, as stated.
Proof version B.
We will only prove the case of formula_127 case is similar.
Define formula_128 which is equivalent to formula_129 and lets us rewrite formula_130, and we have to prove, that formula_131 for some formula_132, which is more intuitive. We further define the set formula_133. Because formula_134 we know, that formula_135 so, that formula_35 is not empty. Moreover, as formula_137, we know that formula_35 is bounded and non-empty, so by Completeness, the supremum formula_139 exists.
There are 3 cases for the value of formula_140, those being formula_141 and formula_131. For contradiction, let us assume, that formula_143. Then, by the definition of continuity, for formula_144, there exists a formula_145 such that formula_116 implies, that formula_147, which is equivalent to formula_148. If we just chose formula_149, where formula_150, then as formula_151, formula_152, from which we get formula_148 and formula_154. It follows that formula_10 is an upper bound for formula_35. However, formula_157, contradicting the upper bound property of the "least upper bound" formula_43, so formula_159. Assume then, that formula_160. We similarly chose formula_161 and know, that there exists a formula_145 such that formula_116 implies formula_164. We can rewrite this as formula_165 which implies, that formula_166. If we now chose formula_167, then formula_166 and formula_169. It follows that formula_10 is an upper bound for formula_35. However, formula_172, which contradict the least property of the "least upper bound" formula_43, which means, that formula_160 is impossible. If we combine both results, we get that formula_131 or formula_20 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 footing.
History.
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:
Let formula_177 be continuous functions on the interval between formula_178 and formula_179 such that formula_180 and formula_181. Then there is an formula_10 between formula_178 and formula_179 such that formula_185.
The equivalence between this formulation and the modern one can be shown by setting formula_186 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.
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" : [0, ∞) → [−1, 1] defined 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 "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 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.
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 cube.
Vrahatis presents a similar generalization to triangles, or more generally, "n"-dimensional simplices. Let "Dn" be an "n"-dimensional simplex with "n"+1 vertices denoted by "v"0...,"vn". Let "F"=("f"1...,"fn") be a continuous function from "Dn" to "Rn", that never equals 0 on the boundary of "Dn". Suppose "F" satisfies the following conditions:
Then there is a point "z" in the interior of "Dn" on which "F"("z")=(0...,0).
It is possible to normalize the "fi" such that "fi"("vi") ;0 for all "i"; then the conditions become simpler:
The theorem can be proved based on the Knaster–Kuratowski–Mazurkiewicz lemma. In can be used for approximations of fixed points and zeros.
General metric and topological spaces.
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 formula_187 and formula_188 are topological spaces, formula_189 is a continuous map, and formula_187 is a connected space, then formula_198 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:
 ;templatestyles src="Math_theorem/styles.css" / ;
Intermediate value theorem ("Version I") — Consider a closed interval formula_13 in the real numbers formula_14 and a continuous function formula_201. Then, if formula_202 is a real number such that formula_203, there exists formula_102 such that formula_126.
The intermediate value theorem is an immediate consequence of these two properties of connectedness:
 ;templatestyles src="Math_proof/styles.css" / ;Proof
By (**), formula_13 is a connected set. It follows from (*) that the image, formula_21, is also connected. For convenience, assume that formula_208. Then once more invoking (**), formula_34 implies that formula_210, or formula_126 for some formula_212. Since formula_213, formula_214 must actually hold, and the desired conclusion follows. The same argument applies if formula_215, so we are done. Q.E.D.
The intermediate value theorem generalizes in a natural way: Suppose that X is a connected topological space and ("Y", ;) is a totally ordered set equipped with the order topology, 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 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.
In constructive mathematics, the intermediate value theorem is not true. Instead, one has to weaken the conclusion:
Practical applications.
A similar result is the Borsuk–Ulam theorem, which says that a continuous map from the formula_225-sphere to Euclidean formula_225-space will always map some pair of antipodal points to the same place.
 ;templatestyles src="Math_proof/styles.css" / ;Proof for 1-dimensional case
Take formula_1 to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points formula_228 and formula_229. Define formula_230 to be formula_231. If the line is rotated 180 degrees, the value −"d" will be obtained instead. Due to the intermediate value theorem there must be some intermediate rotation angle for which "d" = 0, and as a consequence "f"("A") = "f"("B") at this angle.
In general, for any continuous function whose domain is