Intermediate value theorem
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In Mathematical analysis, the intermediate value theorem is either of two theorems of which an account is given below.
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[edit] Intermediate value theorem
The intermediate value theorem states the following: 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)]. I.e.
- f ( I ) ⊇ [f (a), f (b)],
or
- f ( I ) ⊇ [f (b), f (a)].
It is frequently stated in the following equivalent form: Suppose that f : [a, b] → R 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 in (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) = x2-2 from Q to Q satisfies f (0) = -2, f (2) = 2. However there is no rational number x such that f (x) = 0.
[edit] Proof
We shall prove the first case f (a) < u < f (b); the second is similar.
Let S = {x in [a, b] : f(x) ≤ u}. Then S is non-empty (as a is in S) and 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, since f is continuous. But then f (x) > f (c) - ( f (c) - u ) = u whenever | x - c | < δ and then f (x) > u for x in ( c - δ, c + δ) and thus c - δ is an upper bound for S which is smaller than c, a contradiction.
Suppose next 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.
[edit] History
For 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.
[edit] 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.
[edit] 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 reals. Also, let f be bounded (above and below). Then we can say g equals 0 at least once. To see this, consider the following:
Since f is bounded, we can pick a > sup{f(x)} and b < inf{f(x)}. Clearly g(a) < 0 and g(b) > 0. If f is continuous, then g is also continuous. 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 function, x.
[edit] 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 every u between f(a) and f(b) there exists a c between a and b such 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(x) = sin(1/x) for x non-zero, and f(0) = 0. This function is not continuous as the limit for x → 0 does not exist; yet the function has the above intermediate value property.
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).
[edit] Implication 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 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 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.
[edit] 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 the signed area under the function on that interval is equal to the length of the interval b − a multiplied by some function value f(c) such that a < c < b. I.e.,
[edit] 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.
