Continuous function on an interval takes on every value between its values at the ends
Intermediate value theorem: Let be a continuous function defined on and let be a number with . Then there exists some between and such that .
In
mathematical analysis, the intermediate value theorem states that if is a
continuousfunction whose
domain contains the
interval[a, b], then it takes on any given value between and at some point within the interval.
If a continuous function has values of opposite sign inside an interval, then it has a
root in that interval (Bolzano's theorem).[1][2]
The
image of a continuous function over an interval is itself an interval.
Motivation
The intermediate value theorem
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.
Theorem
The intermediate value theorem states the following:
Consider an interval of real numbers and a continuous function . Then
Version I. if is a number between and , that is, then there is a such that .
Version II. the
image set is also a closed interval, and it contains .
Remark:Version II states that the
set of function values has no gap. For any two function values with , even if they are outside the interval between and , all points in the interval are also function values,
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 numbersQ 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.
Proof
Proof version A
The theorem may be proven as a consequence of the
completeness property of the real numbers as follows:[3]
We shall prove the first case, . The second case is similar.
Let be the set of all such that . Then is non-empty since is an element of . Since is non-empty and bounded above by , by completeness, the
supremum exists. That is, is the smallest number that is greater than or equal to every member of .
Note that, due to the continuity of at , we can keep within any of by keeping sufficiently close to . Since is a strict inequality, consider the implication when is the distance between and . No sufficiently close to can then make greater than or equal to , which means there are values greater than in . A more detailed proof goes like this:
Choose . Then such that , Consider the interval . Notice that and every satisfies the condition . Therefore for every we have . Hence cannot be .
Likewise, due to the continuity of at , we can keep within any of by keeping sufficiently close to . Since is a strict inequality, consider the similar implication when is the distance between and . Every sufficiently close to must then make greater than , which means there are values smaller than that are upper bounds of . A more detailed proof goes like this:
Choose . Then such that , Consider the interval . Notice that and every satisfies the condition . Therefore for every we have . Hence cannot be .
With and , it must be the case . Now we claim that .
Fix some . Since is continuous at , such that , .
Since and is open, such that . Set . Then we have
for all . By the properties of the supremum, there exists some that is contained in , and so
Picking , we know that because is the supremum of . This means that
Both inequalities
are valid for all , from which we deduce as the only possible value, as stated.
Proof version B
We will only prove the case of , as the case is similar.[4]
Define which is equivalent to and lets us rewrite as , and we have to prove, that for some , which is more intuitive. We further define the set . Because we know, that so, that is not empty. Moreover, as , we know that is bounded and non-empty, so by Completeness, the
supremum exists.
There are 3 cases for the value of , those being and . For contradiction, let us assume, that . Then, by the definition of continuity, for , there exists a such that implies, that , which is equivalent to . If we just chose , where , then and , so . It follows that is an upper bound for . However, , contradicting the upper bound property of the least upper bound, so . Assume then, that . We similarly chose and know, that there exists a such that implies . We can rewrite this as which implies, that . If we now chose , then and . It follows that is an upper bound for . However, , which contradict the least property of the least upper bound, which means, that is impossible. If we combine both results, we get that or 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[clarification needed] footing.[5]
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.[6] The theorem was first proved by
Bernard Bolzano in 1817. Bolzano used the following formulation of the theorem:[7]
Let be continuous functions on the interval between and such that and . Then there is an between and such that .
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.[8] 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.[9] 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.[10]
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.
Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions;[11] 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[12] 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 v0,...,vn. Let F=(f1,...,fn) 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 fi(vi) is opposite to the sign of fi(x) for all points x on the face opposite to vi;
The sign-vector of f1,...,fn on v0 is not equal to the sign-vector of f1,...,fn on all points on the face opposite to v0.
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:
For all i in 1,...,n, fi(vi)>0, and fi(x)<0 for all points x on the face opposite to vi. In particular, fi(v0)<0.
For all points x on the face opposite to v0, fi(x)>0 for at least one i in 1,...,n.
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:
If and are
metric spaces, is a continuous map, and is a
connected subset, then is connected. (*)
A subset is connected if and only if it satisfies the following property: . (**)
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:
Intermediate value theorem(Version I) — Consider a closed interval in the real numbers and a continuous function . Then, if is a real number such that , there exists such that .
The intermediate value theorem is an immediate consequence of these two properties of connectedness:[14]
Proof
By (**), is a connected set. It follows from (*) that the image, , is also connected. For convenience, assume that . Then once more invoking (**), implies that , or for some . Since , must actually hold, and the desired conclusion follows. The same argument applies if , 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:
Let and be real numbers and be a pointwise continuous function from the
closed interval to the real line, and suppose that and . Then for every positive number there exists a point in the unit interval such that .[15]
Practical applications
A similar result is the
Borsuk–Ulam 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.
Proof for 1-dimensional case
Take to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points and . Define to be . 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 some closed convex -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).[16]
See also
Mean value theorem – On the existence of a tangent to an arc parallel to the line through its endpoints
Non-atomic measure – A measurable set with positive measure that contains no subset of smaller positive measurePages displaying short descriptions of redirect targets
^Essentially follows Clarke, Douglas A. (1971). Foundations of Analysis. Appleton-Century-Crofts. p. 284.
^Slightly modified version of Abbot, Stephen (2015). Understanding Analysis. Springer. p. 123.
^Sanders, Sam (2017). "Nonstandard Analysis and Constructivism!".
arXiv:1704.00281 [
math.LO].
^Bos, Henk J. M. (2001). "The legitimation of geometrical procedures before 1590". Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 23–36.
doi:
10.1007/978-1-4613-0087-8_2.
MR1800805.
^Matthew Frank (July 14, 2020). "Interpolating Between Choices for the Approximate Intermediate Value Theorem". Logical Methods in Computer Science. 16 (3).
arXiv:1701.02227.
doi:
10.23638/LMCS-16(3:5)2020.