Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if

is a continuous function whose domain contains the interval [a, b], then it takes on any given value between

This has two important corollaries: This captures an intuitive property of continuous functions over the real numbers: given

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.

Then Remark: Version II states that the set of function values has no gap.

A subset of the real numbers with no internal gap is an interval.

The theorem depends on, and is equivalent to, the completeness of the real numbers.

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:[3] We shall prove the first case,

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] 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.

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.

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.

However, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false.

Let Dn be an n-dimensional simplex with n+1 vertices denoted by v0,...,vn.

Suppose F satisfies the following conditions: Then there is a point z in the interior of Dn on which F(z)=(0,...,0).

[13] 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

Recall the first version of the intermediate value theorem, stated previously: Intermediate value theorem (Version I) — Consider a closed interval

The intermediate value theorem is an immediate consequence of these two properties of connectedness:[14] By (**),

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.

In constructive mathematics, the intermediate value theorem is not true.

Instead, one has to weaken the conclusion: A similar result is the Borsuk–Ulam theorem, which says that a continuous map from the

Draw a line through the center of the circle, intersecting it at two opposite points

In general, for any continuous function whose domain is some closed convex

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