Intermediate value theorem
The intermediate value theorem says that if a function, , is continuous over a closed interval , and is equal to and at either end of the interval, for any number, c, between and , we can find an so that .
This means that if a continuous function's sign changes in an interval, we can find a root of the function in that interval. For example, if and , we can find an in the interval that is a root of this function, meaning that for this value of x, , if is continuous. This corollary is called Bolzano's theorem.