Encyclopedia > Intermediate value theorem

  Article Content

Intermediate value theorem

Intermediate Value Theorem of Calculus

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.


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>


All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Father Damien

... from which he died at the colony. He is the patron of lepers and outcasts, and was recently made the patron of AIDS patients. The world's only Catholic memorial ...

This page was created in 31.5 ms