Dirichlet-function

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.