From Encyclopedia of Mathematics
Jump to: navigation, search

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:

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. L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098