# Difference between revisions of "Dirichlet-function"

From Encyclopedia of Mathematics

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− | The function which is equal to one at the rational points and to zero at the irrational points. It is also defined by the formula: | + | {{TEX|done}} |

+ | The function $D(x)$ which is equal to one at the rational points and to zero at the irrational points. It is also defined by the formula: | ||

− | + | $$D(x)=\lim_{m\to\infty}\lim_{n\to\infty}(\cos m!\pi x)^{2n},$$ | |

and belongs to the second Baire class (cf. [[Baire classes|Baire classes]]). It is not Riemann integrable on any segment but, since it is equal to zero almost-everywhere, it is Lebesgue integrable. | and belongs to the second Baire class (cf. [[Baire classes|Baire classes]]). It is not Riemann integrable on any segment but, since it is equal to zero almost-everywhere, it is Lebesgue integrable. | ||

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)</TD></TR> | ||

+ | </table> | ||

+ | |||

+ | ====Comment==== | ||

+ | This function is [[Periodic function|periodic]], with any non-zero rational number as [[Period of a function|period]]. |

## Latest revision as of 21:31, 18 November 2017

The function $D(x)$ which is equal to one at the rational points and to zero at the irrational points. It is also defined by the formula:

$$D(x)=\lim_{m\to\infty}\lim_{n\to\infty}(\cos m!\pi x)^{2n},$$

and belongs to the second Baire class (cf. Baire classes). It is not Riemann integrable on any segment but, since it is equal to zero almost-everywhere, it is Lebesgue integrable.

#### References

[1] | I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian) |

#### Comment

This function is periodic, with any non-zero rational number as period.

**How to Cite This Entry:**

Dirichlet-function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dirichlet-function&oldid=12889

This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article