## 6.4. Topology and Continuity

### Examples 6.4.9(b):

Using a computer it is simple enough to draw this function and to see the approximate solution. However, it is even easier to prove that there must be a solution (without specifying where the solution would be).
The function *p(x)* is an odd-degree polynomial. Therefore:

- If
*c =*, then*p(x) =*, so that there exists*A*such that*p(A) > 0* - If
*c = -*, then*p(x) = -*, so that there exists*B*such that*p(B) < 0*

Hence, by Bolzano's theorem there exists a zero of *p(x)* between
the (unknown !) numbers *A* and *B*.