Intermediate value theorem

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 non-empty (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.

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.

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>b-a</math> multiplied by some function value <math>f(c)</math> such that <math>a < c < b</math>.

More specifically:

<math>f(x) \in [a,b] \Rightarrow \exists c</math>

such that

<math>\int_a^b f(x) dx = (b-a) f(c)</math>

holds.