In mathematics, the Dirichlet function is the indicator function ${\displaystyle \mathbf {1} _{\mathbb {Q} }}$ of the set of rational numbers ${\displaystyle \mathbb {Q} }$ over the set of real numbers ${\displaystyle \mathbb {R} }$, i.e. ${\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=1}$ for a real number x 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. 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 ${\displaystyle \mathbb {R} }$.

Using an enumeration of the rational numbers between 0 and 1, we define the function f_n (for all nonnegative integer n) as the indicator function of the set of the first n terms of this sequence of rational numbers. The increasing sequence of functions f_n (which are nonnegative, Riemann-integrable with a vanishing integral) pointwise converges to the Dirichlet function which is not Riemann-integrable.