Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value.

Intermediate value theorem

Intermediate Value Theorem

• Version I. The intermediate value theorem states the following: If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u.

• Version II. Suppose that I is an interval [a, b] in the real numbers R and that f : I → R is a continuous function. Then the image set f(I) is also an interval, and either it contains [f(a), f(b)], or it contains [f(b), f(a)]; that is,

f(I) ⊇ [f(a), f(b)], or f(I) ⊇ [f(b), f(a)].

It is frequently stated in the following equivalent form: Suppose that f : [a, b] → R is continuous and that u is a real number satisfying f(a) < u < f(b) or f(a) > u > f(b). Then for some c ∈ [a, b], f(c) = u.

This captures an intuitive property of continuous functions: given f continuous on [1, 2], if f(1) = 3 and f(2) = 5 then f must take the value 4 somewhere between 1 and 2. It represents the idea that the graph of a continuous function on a closed interval can only be drawn without lifting your pencil from the paper.

The theorem depends on (and is actually equivalent to) the completeness of the real numbers. It is false for the rational numbers Q. For example, the function f(x) = x² − 2 for x ∈ Q satisfies f(0) = −2 and f(2) = 2. However there is no rational number x such that f(x) = 0, because if so, then √2 would be rational.

Proof

We shall prove the first case f(a) < u < f(b); the second is similar.

Let S be the set of all x in [a, b] such that f(x) ≤ u. Then S is non-empty since a is an element of S, and S is bounded above by b. Hence, by the completeness property of the real numbers, the supremum c = sup S exists. That is, c is the lowest number that is greater than or equal to every member of S. We claim that f(c) = u.

• Suppose first that f(c) > u, then f(c) − u > 0. Since f is continuous, there is a δ > 0 such that | f(x) − f(c) | < ε whenever | x − c | < δ. Pick ε = f(c) − u, then | f(x) − f(c) | < f(c) − u. But then f(x) > f(c) − (f(c) − u) = u whenever | x − c | < δ (that is, f(x) > u for x in (c − δ, c + δ)). Thus c − δ is an upper bound for S, a contradiction since we assumed that c was the least upper bound and c − δ < c.

• Suppose instead that f(c) < u. Again, by continuity, there is a δ > 0 such that | f(x) − f(c) | < u − f(c) whenever | x − c | < δ. Then f(x) < f(c) + (u − f(c)) = u for x in (c − δ, c + δ) and there are numbers x greater than c for which f(x) < u, again a contradiction to the definition of c.

We deduce that f(c) = u as stated.

An alternative proof may be found at non-standard calculus.

History

For u = 0 above, the statement is also known as Bolzano's theorem. This theorem was first proved by Bernard Bolzano in 1817. In 1817, Cauchy provided a proof in 1821. Inspired both by the goal of formalizing the analysis of functions and the work of Lagrange. The idea that continuous functions posses the intermediate value theorem has a more early origin. Before the formal definition of continuity was given the intermediate value property was given as part of the definition of what a continuous function should be. For example F. A Arbogast (who won the St. Pertersburg Academy's prize competition in 1787 for giving the best characterization of the functions that are solution of the vibrating string equation) assumed the functions to have no jumps, satisfy the intermediate value theorem and have increments who's sizes correspond to the sizes of the increments of the variable. As examples show, there are non-continuous functions having the intermediate value theorem (even nowhere continuous functions). 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.

Generalization

The intermediate value theorem can be seen as a consequence of the following two statements from topology:

• If X and Y are topological spaces, f : X → Y is continuous, and X is connected, then f(X) is connected.
• A subset of R is connected if and only if it is an interval.

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.

Converse is false

Suppose f is a real-valued function defined on some interval I, and for every two elements a and b in I and for all u in the open interval bounded by f(a) and f(b) there is a c in the open interval bounded by a and b so that f(c) = u. Does f have to be continuous? The answer is no; the converse of the intermediate value theorem fails.

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 above 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; this definition was not adopted.

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).

Implications of theorem in real world

The theorem implies that on any great circle around the world, the temperature, pressure, elevation, carbon dioxide concentration, or any other similar quantity which varies continuously, there will always exist two antipodal points that share the same value for that variable.

Proof: Take f to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points A and B. Let d be defined by the difference f(A) − f(B). 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.

This is a special case of a more general result called the Borsuk–Ulam theorem.

The theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily-met constraints).

See also

• Mean value theorem
• Darboux's theorem
• Borsuk-Ulam theorem
• Hairy ball theorem

External links

• Intermediate value Theorem - Bolzano Theorem at cut-the-knot
• Bolzano's Theorem by Julio Cesar de la Yncera, Wolfram Demonstrations Project.
• Weisstein, Eric W. "Bolzano's Theorem". MathWorld.