Dirichlet function

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

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, ${\mathbf{𝟏}}_{\mathbb{\mathbb{Q}}}(x+T)={\mathbf{𝟏}}_{\mathbb{\mathbb{Q}}}(x)$. 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 $\mathbb{ℝ}$.

See also

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