Intermediate value theorem

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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

File:Intermediatevaluetheorem.png

Intermediate Value Theorem

Version I. The intermediate value theorem states the following: If the function y = f(x) is continuous 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,

Template:Nowrap or Template:Nowrap

It is frequently stated in the following equivalent form: Suppose that Template:Nowrap is continuous and that u is a real number satisfying Template:Nowrap or Template:Nowrap 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 Template:Nowrap 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 Template:Nowrap 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 stated by Bernard Bolzano (1781–1848) in 1817, together with a proof which used techniques which were especially rigorous for their time but which are now regarded as non-rigorous.^[1]

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.

Example of use in proof

The theorem is rarely applied with concrete values (though it can be and is used for showing that functions have inverses); instead, it gives some characterization of continuous functions. For example, let g(x) = f(x) − x for f continuous over the real numbers. Also, let f be bounded (above and below). Then we can say g = 0 at least once. To see this, consider the following:

Since f is bounded, we can pick a greater than Template:Nowrap and b less than Template:Nowrap Clearly g(a) < 0 and g(b) > 0. f is continuous, then g is also continuous by the continuity of the subtraction operation. Since g is continuous, we can apply the intermediate value theorem and state that g must take on the value of 0 somewhere between a and b. This result proves that any continuous bounded function must cross the identity function id(x) = x and thus has a fixed point.

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).^[1]