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In mathematics, the Dirichlet function is the indicator function 1ℚ of the set of rational numbers ℚ, i.e. 1ℚ (x) = 1 if x is a rational number and 1ℚ (x) = 0 if x is not a rational number (i.e. an irrational number).

It is named after the mathematician Peter Gustav Lejeune Dirichlet. It is an example of pathological function which provides counterexamples to many situations.

Topological properties
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 * The Dirichlet function is nowhere continuous.
 * Its restriction to the set of rational numbers and irrational numbers are constants and therefore continuous. The Dirichlet function is an archetypal example of Blumberg theorem.


 * The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
 * $$f(x)=\lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)\right)^{2j}\right)$$
 * for integer j and k. This shows that the Dirichlet function is a Baire class 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a meagre set.

Integration properties
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 * 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 (for the Lebesgue measure).
 * 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 (for the Lebesgue measure).