Template:Short description In mathematics, the Dirichlet function<ref>Template:Springer</ref><ref>Dirichlet Function — from MathWorld</ref> is the indicator function <math>\mathbf{1}_\Q</math> of the set of rational numbers <math>\Q</math>, i.e. <math>\mathbf{1}_\Q(x) = 1</math> if Template:Mvar is a rational number and <math>\mathbf{1}_\Q(x) = 0</math> if Template:Mvar is not a rational number (i.e. is an irrational number). <math display="block">\mathbf 1_\Q(x) = \begin{cases} 1 & x \in \Q \\ 0 & x \notin \Q \end{cases}</math>
It is named after the mathematician Peter Gustav Lejeune Dirichlet.<ref>Template:Cite journal</ref> It is an example of a pathological function which provides counterexamples to many situations.
Topological propertiesEdit
PeriodicityEdit
For any real number Template:Mvar and any positive rational number Template:Mvar, <math>\mathbf{1}_\Q(x + T) = \mathbf{1}_\Q(x)</math>. 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 <math>\R</math>.
Integration propertiesEdit
See alsoEdit
- Thomae's function, a variation that is discontinuous only at the rational numbers