Dirichlet function

Template:Short description In mathematics, the Dirichlet function is the indicator function ${\mathrm{𝟏}}_{\mathbb{Q}}$ of the set of rational numbers $\mathbb{Q}$, i.e. ${\mathrm{𝟏}}_{\mathbb{Q}}(x)=1$ if Template:Mvar is a rational number and ${\mathrm{𝟏}}_{\mathbb{Q}}(x)=0$ if Template:Mvar is not a rational number (i.e. is an irrational number). $${\mathrm{𝟏}}_{\mathbb{Q}}(x)=\{\begin{array}{cc}1& x\in \mathbb{Q}\\ 0& x\notin \mathbb{Q}\end{array}$$

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

Periodicity

For any real number Template:Mvar and any positive rational number Template:Mvar, ${\mathrm{𝟏}}_{\mathbb{Q}}(x+T)={\mathrm{𝟏}}_{\mathbb{Q}}(x)$. 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