# example of a non Riemann integrable function

Let $[a,b]$ be any closed interval^{} and
consider the Dirichlet’s function $f:[a,b]\to \mathbb{R}$

$$f(x)=\{\begin{array}{cc}1\hfill & \text{if}x\text{is rational}\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$ |

Then $f$ is not Riemann integrable^{}. In fact given any interval $[{x}_{1},{x}_{2}]\subset [a,b]$ with $$ one has

$$\underset{[{x}_{1},{x}_{2}]}{sup}f(x)=1,\underset{[{x}_{1},{x}_{2}]}{inf}f(x)=0$$ |

because every interval contains both rational and irrational points. So all upper Riemann sums are equal to $1$ and all lower Riemann sums are equal to $0$.

Title | example of a non Riemann integrable function |
---|---|

Canonical name | ExampleOfANonRiemannIntegrableFunction |

Date of creation | 2013-03-22 15:03:28 |

Last modified on | 2013-03-22 15:03:28 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 4 |

Author | paolini (1187) |

Entry type | Example |

Classification | msc 28-XX |

Classification | msc 26-XX |