# Difference between revisions of "Intermediate Value Theorem"

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==Statement== | ==Statement== | ||

− | + | Take a function <math>f</math> and interval <math>[a,b]</math> such that the following hold: | |

− | + | <math>f:[a,b]\rightarrow\mathbb{R},</math> | |

− | + | <math>f</math> is continuous on <math>[a,b],</math> | |

− | Then, <math>\exists c\in (a,b)</math> such that <math>f(c)=k</math> | + | <math>f(a)<k<f(b).</math> |

+ | |||

+ | Then, <math>\exists c\in (a,b)</math> such that <math>f(c)=k.</math> | ||

==Proof== | ==Proof== | ||

− | Consider <math>g:[a,b]\rightarrow\mathbb{R}</math> such that <math>g(x)=f(x)-k | + | Consider <math>g:[a,b]\rightarrow\mathbb{R}</math> such that <math>g(x)=f(x)-k.</math> |

− | |||

− | |||

− | + | Note that <math>g(a)<0</math> and <math>g(b)>0</math> | |

− | or <math>f(c)=k</math> | + | By the [[Location of roots theorem]], <math>\exists c\in (a,b)</math> such that <math>g(c)=0</math> or <math>f(c)=k.</math> |

<p align=right>QED</p> | <p align=right>QED</p> | ||

## Latest revision as of 18:08, 1 February 2021

The **Intermediate Value Theorem** is one of the very interesting properties of continous functions.

## Statement

Take a function and interval such that the following hold:

is continuous on

Then, such that

## Proof

Consider such that

Note that and

By the Location of roots theorem, such that or

QED