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2 Intermediate value theorem of integration |

The **intermediate value theorem** of 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. For u=0 above, it is also known as *Bolzano's theorem* and follows immediately from the intermediate value theorem of calculus.
This theorem was first stated, together with a proof which used techniques which are now regarded as non-rigorous, by Bernard Bolzano.

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.

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 is a continuous function on some interval (which is typically the real numbers, **R**). Then the area under the function on a certain interval is equal to the length of the interval multiplied by some function value such that .