# 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 |