# Dirichlet-function

From Encyclopedia of Mathematics

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://www.encyclopediaofmath.org/index.php?title=Dirichlet-function&oldid=42322

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