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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:

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

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.


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