NationMaster - Encyclopedia: Intermediate value theorem

In Mathematical analysis, the intermediate value theorem is either of two theorems of which an account is given below. Analysis has its beginnings in the rigorous formulation of calculus. ...

1 Intermediate value theorem
2 Intermediate value theorem of integration
3 Intermediate value theorem of derivatives
4 External links

Intermediate value theorem

Intermediate Value Theorem

The intermediate value theorem states the following: Suppose that $displaystyle I$ is an interval in the real numbers and that $fcolon I rightarrow mathbb{R}$ is a continuous function. Then the image set $displaystyle f(I)$ is also an interval, and either it contains $displaystyle [f(a),f(b)]$ , or it contains $displaystyle [f(b),f(a)]$ . I.e. Intermediate Value Theorem File links The following pages link to this file: Intermediate value theorem Categories: GFDL images ... Intermediate Value Theorem File links The following pages link to this file: Intermediate value theorem Categories: GFDL images ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

$displaystyle f(I) supseteq [f(a),f(b)]$ ,

$displaystyle f(I) supseteq [f(b),f(a)]$ .

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

This captures an intuitive property of continuous functions: given continuous on $displaystyle [1,2]$ , if $displaystyle f(1) = 3$ and $displaystyle f(2) = 5$ then 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 $mathbb{Q}$ . For example, the function $displaystyle f(x) = x^{2} - 2, , x inmathbb{Q}$ satisfies $f(0) = -2,, f(2) = 2$ . However there is no rational number $displaystyle x$ such that $displaystyle f(x) = 0$ . In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

Proof

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

Let $displaystyle text{S} = { x in [a,b] : f(x) leq u }$ . Then $displaystyle S$ is non-empty (since $displaystyle a in displaystyle S$ ) and bounded above by $displaystyle b$ . Hence by the completeness property of the real numbers, the supremum $c = sup text{S}$ exists. We claim that $displaystyle f(c) = u$ . In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

Suppose first that $displaystyle f(c) > u$ . Then $displaystyle f(c) - u > 0$ , so there is a $displaystyle delta > 0$ such that $displaystyle |f(x) - f(c)| < f(c) - u$ whenever $displaystyle |x - c| < delta$ , since is continuous. But then $displaystyle f(x) > f(c) - (f(c) - u) = u$ whenever $displaystyle |x - c| < delta$ (i.e. $displaystyle f(x) > u$ for $displaystyle x$ in $displaystyle(c - delta, c + delta)$ ). Thus $displaystyle c - delta$ is an upper bound for $displaystyle S$ , a contradiction since we assumed that $displaystyle c$ was the upper bound and $displaystyle c - delta < displaystyle c$ .

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

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

History

For $displaystyle u = 0$ 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. Bernard Bolzano Bernard (Bernhard) Placidus Johann Nepomuk Bolzano (October 5, 1781 â€“ December 18, 1848) was a Bohemian mathematician, theologian, philosopher, logician and antimilitarist of German mother tongue. ...

Generalization

The intermediate value theorem can be seen as a consequence of the following two statements from topology: A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...

If $displaystyle X$ and $displaystyle Y$ are topological spaces, $fcolon X rightarrow Y$ is continuous, and $displaystyle X$ is connected, then $displaystyle f(X)$ is connected.
A subset of is connected if and only if it is an interval.

Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...

Example of use in proof

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

Since is bounded, we can pick $a > sup(f(x))$ and $b < inf(f(x))$ . Clearly $displaystyle g(a) < 0$ and $displaystyle g(b) > 0$ . If is continuous, then $displaystyle g$ is also continuous. Since $displaystyle g$ is continuous, we can apply the intermediate value theorem and state that $displaystyle g$ must take on the value of 0 somewhere between $displaystyle a$ and $displaystyle b$ . This result proves that any continuous bounded function must cross the function, $displaystyle x$ .

Converse is false

Suppose is a real-valued function defined on some interval $displaystyle I$ , and for every two elements $displaystyle a$ and $displaystyle b$ in $displaystyle I$ and $forall, u in (f(a),f(b)), exists , c in (a,b)$ such that $displaystyle f(c) = u$ . Does have to be continuous? The answer is no; the converse of the intermediate value theorem fails. As an example, take the function $displaystyle f(x) = sin(1/x)$ for $x neq 0$ , and $displaystyle f(0) = 0$ . This function is not continuous as the limit when $displaystyle x$ 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. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... 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. ...

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). This article is about Darbouxs theorem in real analysis. ... For a non-technical overview of the subject, see Calculus. ... In mathematics, a Darboux function, named for Gaston Darboux (1842-1917), is a real-valued function f which has the intermediate value property: on the interval between a and b, f assumes every real value between f(a) and f(b). ...

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. For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the... For other uses, see Temperature (disambiguation). ... This article is about pressure in the physical sciences. ... Elevation histogram of the surface of the Earth â€“ approximately 71% of the Earths surface is covered with water. ... In order to meet Wikipedias quality standards, this article requires cleanup. ... Antipodal points on the surface of a sphere are diametrically opposite; on the other side of a globe. ...

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 $displaystyle A$ and $displaystyle B$ . Let $displaystyle d$ be the difference $displaystyle f(A)-f(B)$ . If the line is rotated 180 degrees, the value $displaystyle -d$ 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 Borsukâ€“Ulam theorem states that any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. ...

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: The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ...

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., In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ...

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.

External links

Intermediate value Theorem - Bolzano Theorem at cut-the-knot

Categories: Continuous mappings | Mathematical theorems | Articles containing proofs cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

Results from FactBites:

Intermediate value theorem - Wikipedia, the free encyclopedia (797 words)
	The intermediate value theorem states the following: Suppose that I is an interval in the real numbers R and that f : I → R is a continuous function.
	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).
	The intermediate value theorem of integration is derived from the mean value theorem and states:

Encyclopedia: Intermediate value theorem (1346 words)
	The intermediate value theorem can be seen as a consequence of the following two statements from topology: Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces.
	Darbouxs theorem is a theorem in symplectic topology which states that every symplectic manifold (of fixed dimension) is locally symplectomorphic.
	The intermediate value theorem of integration is derived from the mean value theorem and states: Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc....

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