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 $f$ is continuous on a closed interval $I$ and the values of $f$ in the end points of $I$ 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  $f(x)=0$.