Dirichlet function

In mathematics, the Dirichlet function[1][2] is the indicator function 1_ℚ of the set of rational numbers ℚ, i.e. 1_ℚ(x) = 1 if x is a rational number and 1_ℚ(x) = 0 if x is not a rational number (i.e. an irrational number).

It is named after the mathematician Peter Gustav Lejeune Dirichlet.[3] It is an example of pathological function which provides counterexamples to many situations.

Topological properties

• The Dirichlet function is nowhere continuous.

Proof —

• If y is rational, then f(y) = 1. To show the function is not continuous at y, we need to find an ε such that no matter how small we choose δ, there will be points z within δ of y such that f(z) is not within ε of f(y) = 1. In fact, 1/2 is such an ε. Because the irrational numbers are dense in the reals, no matter what δ we choose we can always find an irrational z within δ of y, and f(z) = 0 is at least 1/2 away from 1.
• If y is irrational, then f(y) = 0. Again, we can take ε = 1/2, and this time, because the rational numbers are dense in the reals, we can pick z to be a rational number as close to y as is required. Again, f(z) = 1 is more than 1/2 away from f(y) = 0.

Its restrictions to the set of rational numbers and to the set of 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:

${\displaystyle \forall x\in \mathbb {R} ,\quad \mathbf {1} _{\mathbb {Q} }(x)=\lim _{k\to \infty }\left(\lim _{j\to \infty }\left(\cos(k!\pi x)\right)^{2j}\right)}$

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.[4]

Periodicity

For any real number x and any positive rational number T, 1_ℚ(x + T) = 1_ℚ(x). The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods, the set of rational numbers, is a dense subset of ℝ.

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 (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.

Proof —

Using an enumeration of the rational numbers between 0 and 1, we define the function f_n(for all nonnegative integer n) as the indicator function of the set of the first n terms of this sequence of rational numbers. The increasing sequence of functions f_n (which are nonnegative, Riemann-integrable with a vanishing integral) pointwise converges to the Dirichlet function which is not Riemann-integrable.

• 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).

References

1. "Dirichlet-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
2. Dirichlet Function — from MathWorld
3. Lejeune Dirichlet, Peter Gustav (1829). "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données". Journal für die reine und angewandte Mathematik. 4: 157–169.
4. Dunham, William (2005). The Calculus Gallery. Princeton University Press. p. 197. ISBN 0-691-09565-5.