## 6.4. Topology and Continuity

### Theorem 6.4.10: Intermediate Value Theorem

If

*f*is continuous on a closed interval*[a, b]*and*d*is any number between*f(a)*and*f(b)*. Then there exists a number*c*in the open interval*(a, b)*such that*f(c) = d*.### Proof:

With the work we have done so far this proof is easy. In fact, the easiest proof is an application of Bolzano's theorem, and is left as an exercise.