## 6.4. Topology and Continuity

### Theorem 6.4.8: Bolzano Theorem

If

*f*is continuous on a closed interval*[a, b]*and*f(a)*and*f(b)*have opposite signs, then there exits a number*c*in the open interval*(a, b)*such that*f(c) = 0*.### Proof:

With the work we have done so far this proof is easy. Since *(a, b)* is
connected and *f* a continuous function, the interval
*f( (a,b) )* is also connected. Therefore that set must contain the interval
*(f(a), f(b))* (assuming that *f(a) < f(b)* ). Since *f(a)*
and *f(b)* have opposite signs, this interval must include *0*. Therefore,
*0* is in the image of *(a, b)*, or equivalently: there exists a *c*
in open interval *(a, b)* such that *f(c) = 0*.

That's it !