# intermediate value theorem

In Mathematical analysis, the intermediate value theorem is either of two theorems of which an account is given below.

## Intermediate value theorem

Intermediate Value Theorem
The intermediate value theorem states the following: If y=f(x) is continuous on [a,b], and N is a number between f(a) and f(b), then there is at least 1 c ∈ [a,b] such that f(c) = N.

Suppose that is an interval [a,b] in the real numbers R and that is a continuous function. Then the image set is also an interval, and either it contains [f(a),f(b)], or it contains [f(b),f(a)]; that is,
• ,
or
• .
It is frequently stated in the following equivalent form: Suppose that 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 be equal to 4 somewhere between 1 and 2. It represents the idea that the graph of a continuous function can be drawn without lifting your pencil from the paper.

The theorem depends on the completeness of the real numbers. It is false for the rational numbers Q. For example, the function f(x) = x^2 - 2, xQ satisfies . However there is no rational number such that .

### Proof

We shall prove the first case ; the second is similar.

Let . Then is non-empty (since ) and bounded above by . Hence by the completeness property of the real numbers, the supremum exists. We claim that .

Suppose first that . Then , so there is a such that whenever , since is continuous. But then whenever (i.e. for in ). Thus is an upper bound for , a contradiction since we assumed that was the least upper bound and .

Suppose next that . Again, by continuity, there is a such that whenever . Then for in and there are numbers greater than for which , again a contradiction to the definition of .

We deduce that as stated.

### History

For above, the statement is also known as Bolzano's theorem; this theorem was first stated by Bernard Bolzano, together with a proof which used techniques which were especially rigorous for their time but which are now regarded as non-rigorous.

### Generalization

The intermediate value theorem can be seen as a consequence of the following two statements from topology:
• If and are topological spaces, is continuous, and is connected, then is connected.
• A subset of is connected if and only if it is an interval.

### Example of use in proof

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

Since is bounded, we can pick and . Clearly and . If is continuous, then is also continuous. Since is continuous, we can apply the intermediate value theorem and state that must take on the value of 0 somewhere between and . This result proves that any continuous bounded function must cross the function, .

### Converse is false

Suppose is a real-valued function defined on some interval , and for every two elements and in and such that . Does have to be continuous? The answer is no; the converse of the intermediate value theorem fails. As an example, take the function for , and . This function is not continuous as the limit when gets close 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 anything else that varies continuously, there will always exist two antipodal points that share the same value for that variable.

Proof: 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 . Let be the difference . If the line is rotated 180 degrees, the value will be obtained instead. Due to the intermediate value theorem there must be some intermediate rotation angle for which , and as a consequence 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]

## Intermediate value theorem of integration

The intermediate value theorem of integration is derived from the mean value theorem and states:

If is a continuous function on some interval , then the signed area under the function on that interval is equal to the length of the interval multiplied by some function value such that . I.e.,

## Intermediate value theorem of derivatives

If is a differentiable real-valued function on , then the (first order) derivative has the intermediate value property, though might not be continuous.

Analysis has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function.
In algebra, an interval is a set that contains every real number between two indicated numbers and may contain the two numbers themselves. Interval notation is the notation in which permitted values for a variable are expressed as ranging over a certain interval; "" is an
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction , where b is not zero.
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Bernard (Bernhard) Placidus Johann Nepomuk Bolzano (September 5 1781 – December 18, 1848) was a Bohemian mathematician, theologian, philosopher, logician and antimilitarist of German mother tongue.
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The Conway base 13 function is a function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, even though Conway's function f is not continuous, if f(a
Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions which result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval.
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A great circle
trillion fold).]]

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