Dirichlet function

Topological properties

• The Dirichlet function is nowhere continuous.
• The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:

  Periodicity

Integration properties

• The Dirichlet function is not Riemann-integrable on any segment of ℝ whereas it is bounded because the set of its discontinuity points is not negligible.
• The Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral.

• The Dirichlet function is Lebesgue-integrable on ℝ and its integral over ℝ is zero because it is zero except on the set of rational numbers which is negligible.