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  $f(x)=0$.

Title	Bolzano’s theorem
Canonical name	BolzanosTheorem
Date of creation	2013-03-22 15:39:06
Last modified on	2013-03-22 15:39:06
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	5
Author	pahio (2872)
Entry type	Theorem
Classification	msc 26A06
Related topic	PolynomialEquationOfOddDegree
Related topic	Evolute2
Related topic	ExampleOfConvergingIncreasingSequence