# 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: | 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: | ||

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

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. |

## Revision as of 14:24, 28 August 2014

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:

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

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) |

**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