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

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.

For any real number x and any positive rational number T, ${\displaystyle \mathbf {1} _{\mathbb {Q} }(x+T)=\mathbf {1} _{\mathbb {Q} }(x)}$. 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 ${\displaystyle \mathbb {R} }$.

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