Dirichlet function explained

1 _{\Q} of the set of rational numbers \Q , i.e. 1 _{\Q(x)} = 1 if is a rational number and 1 _{\Q(x)} = 0 if is not a rational number (i.e. is an irrational number).

It is named after the mathematician Peter Gustav Lejeune Dirichlet.^[2] It is an example of a pathological function which provides counterexamples to many situations.

Periodicity

For any real number and any positive rational number,

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 .

See also

• Thomae's function, a variation that is discontinuous only at the rational numbers

Notes and References

1. http://mathworld.wolfram.com/DirichletFunction.html Dirichlet Function — from MathWorld
2. Peter Gustav . Lejeune Dirichlet . 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 . 1829 . 157–169.