Dirichlet function

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.