- Intermediate value theorem
In
mathematical analysis , the intermediate value theorem states that for each value between the upper and lower bounds of the image of acontinuous function there is a corresponding value in its domain mapping to the original.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 number s 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,:: nowrap|"f"("I") ⊇ ["f"("a"), "f"("b")] , or nowrap|"f"("I") ⊇ ["f"("b"), "f"("a")] .It is frequently stated in the following equivalent form: Suppose that nowrap|"f" : ["a", "b"] → R is continuous and that "u" is a real number satisfying nowrap|"f"("a") < "u" < "f"("b") or nowrap|"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 the completeness of the
real number s. It is false for therational number s Q. For example, the function nowrap|1="f"("x") = "x"2 − 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 nowrap|"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. We claim that "f"("c") = "u".* Suppose first that "f"("c") > "u". Then "f"("c") − "u" > 0, so there is a "δ" > 0 such that | "f"("x") − "f"("c") | < "f"("c") − "u" whenever | "x" − "c" | < "δ" because "f" is continuous. 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), 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 "X" and "Y" aretopological space s, "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; 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 nowrap|sup{"f"("x") : "x" ∈ R} and "b" less than nowrap|inf{"f"("x") : "x" ∈ R}. 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".
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, thetemperature ,pressure ,elevation ,carbon dioxide concentration, or anything other similar quantity which varies continuously, there will always exist twoantipodal 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). [
Keith Devlin (2007) [http://www.maa.org/devlin/devlin_02_07.html How to stabilize a wobbly table] ]Intermediate value theorem of integration
The intermediate value theorem of integration is derived from the
mean value theorem and states:If "f" is a continuous function on some interval ["a", "b"] , then there exists a "c" with "a" < "c" < "b" such that the signed area under the function on that interval is equal to the length of the interval "b" − "a" multiplied by "f"("c"). That is, :
Intermediate value theorem of derivatives
If "f" is a differentiable real-valued function on R, then the (first order) derivative "f"′ has the intermediate value property, though "f"′ might not be continuous.
References
External links
* [http://www.cut-the-knot.org/Generalization/ivt.shtml Intermediate value Theorem - Bolzano Theorem] at
cut-the-knot
* [http://demonstrations.wolfram.com/BolzanosTheorem/ Bolzano's Theorem] by Julio Cesar de la Yncera,The Wolfram Demonstrations Project .
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