Bolzano’s theorem
A continuous function can not change its sign (http://planetmath.org/SignumFunction) without going through the zero.
This contents of Bolzano’s theorem may be formulated more precisely as the
Theorem.
If a real function is continuous on a closed interval and the values of in the end points of have opposite (http://planetmath.org/Positive) signs, then there exists a zero of this function inside the interval.
The theorem is used when using the interval halving method for getting an approximate value of a root of an equation of the form .