# intermediate value theorem

If $f$ is a real-valued continuous function^{} on the interval $[a,b]$,
and ${x}_{1}$ and ${x}_{2}$ are points with $$
such that $f({x}_{1})\ne f({x}_{2})$,
then for every $y$ strictly between $f({x}_{1})$ and $f({x}_{2})$
there is a $c\in ({x}_{1},{x}_{2})$ such that $f(c)=y$.

Bolzano’s theorem is a special case of this.

The theorem can be generalized as follows:
If $f$ is a real-valued continuous function
on a connected^{} topological space^{} $X$,
and ${x}_{1},{x}_{2}\in X$ with $f({x}_{1})\ne f({x}_{2})$,
then for every $y$ between $f({x}_{1})$ and $f({x}_{2})$
there is a $\xi \in X$ such that $f(\xi )=y$.
(However, this “generalization^{}” is essentially trivial,
and in order to derive the intermediate value theorem from it
one must first establish the less trivial fact that $[a,b]$ is connnected.)
This result remains true
if the codomain is an arbitrary ordered set with its order topology;
see the entry
proof of generalized intermediate value theorem (http://planetmath.org/ProofOfGeneralizedIntermediateValueTheorem)
for a proof.

Title | intermediate value theorem |

Canonical name | IntermediateValueTheorem |

Date of creation | 2013-03-22 11:51:29 |

Last modified on | 2013-03-22 11:51:29 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 15 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 26A06 |

Classification | msc 70F25 |

Classification | msc 17B50 |

Classification | msc 81-00 |

Related topic | RollesTheorem |

Related topic | MeanValueTheorem |

Related topic | Continuous |