Dirichlet function

In mathematics, the Dirichlet function^[1][2] is the indicator function ${\displaystyle {\mathbf{𝟏}}_{\mathbb{\mathbb{Q}}}}$ of the set of rational numbers ${\displaystyle \mathbb{\mathbb{Q}}}$ over the set of real numbers ${\displaystyle \mathbb{ℝ}}$, i.e. ${\displaystyle {\mathbf{𝟏}}_{\mathbb{\mathbb{Q}}}(x)=1}$ for a real number x if x is a rational number and ${\displaystyle {\mathbf{𝟏}}_{\mathbb{\mathbb{Q}}}(x)=0}$ if x is not a rational number (i.e. is an irrational number). $${\displaystyle {\mathbf{𝟏}}_{\mathbb{\mathbb{Q}}}(x)=\{\begin{array}{ll}1& x\in \mathbb{\mathbb{Q}}\\ 0& x\notin \mathbb{\mathbb{Q}}\end{array}}$$

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