The Intermediate value theorem in calculus states the following: Suppose that I is an interval in the real numbers R and that f : I > R is a continuous function. Then the image set f ( I ) is also an interval.
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: 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.
Proof of the theorem: 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 nonempty (as a is in S) and bounded above by b. Hence by the continuum 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 an 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.
The intermediate value theorem is in essence equivalent to Rolle's theorem.
The intermediate value theorem can be seen as a consequence of the following two statements from topology:
Intermediate Value Theorem of Integration
In integration the intermediate value theorem has a different twist. In this context (derived from the intermediate value theorem above) it is used to refer to the following fact:
Assume <math>f</math> is a continuous function on some interval <math>I</math> (which is typically the real numbers, R). Then the area under the function in a certain interval <math>[a,b]</math> is equal to the length of the interval <math>ba</math> multiplied by some function value <math>f(c)</math> such that <math>a < c < b</math>.
More specifically:
such that
holds.
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