# function of not bounded variation

Example. We show that the function

$f:x\mapsto $ | $\{\begin{array}{cc}x\mathrm{cos}{\displaystyle \frac{\pi}{x}}\hfill & \text{when}x\ne 0,\hfill \\ 0\hfill & \text{when}x=0,\hfill \end{array}$ |

which is continuous^{} in the whole $\mathbb{R}$, is not of bounded variation^{} on any interval containing the zero.

Let us take e.g. the interval $[0,a]$. Chose a positive integer $m$ such that $$ and the partition of the interval with the points $\frac{1}{m},\frac{1}{m+1},\frac{1}{m+2},\mathrm{\dots},\frac{1}{n}$ into the subintervals $[0,\frac{1}{n}],[\frac{1}{n},\frac{1}{n-1}],\mathrm{\dots},[\frac{1}{m+1},\frac{1}{m}],[\frac{1}{m},a]$. For each positive integer $\nu $ we have (see this (http://planetmath.org/CosineAtMultiplesOfStraightAngle))

$$f\left(\frac{1}{\nu}\right)=\frac{1}{\nu}\mathrm{cos}\nu \pi =\frac{{(-1)}^{\nu}}{\nu}.$$ |

Thus we see that the total variation^{} of $f$ in all partitions of $[0,a]$ is at least

$$\frac{1}{n}+\left(\frac{1}{n}+\frac{1}{n-1}\right)+\mathrm{\dots}+\left(\frac{1}{m+1}+\frac{1}{m}\right)=\frac{1}{m}+2\sum _{\nu =m+1}^{n}\frac{1}{\nu}.$$ |

Since the harmonic series diverges, the above sum increases to $\mathrm{\infty}$ as $n\to \mathrm{\infty}$. Accordingly, the total variation must be infinite, and the function $f$ is not of bounded variation on $[0,a]$.

It is not difficult to justify that $f$ is of bounded variation on any finite interval that does not contain 0.

## References

- 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III. Toinen osa. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).

Title | function of not bounded variation |
---|---|

Canonical name | FunctionOfNotBoundedVariation |

Date of creation | 2013-03-22 17:56:29 |

Last modified on | 2013-03-22 17:56:29 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Example |

Classification | msc 26A45 |

Synonym | example of unbounded variation |

Synonym | function of unbounded variation |