Dirichlet function

In mathematics, the Dirichlet function is the indicator function $\mathbf {1} _{\mathbb {Q} }$ of the set of rational numbers $\mathbb {Q}$ , i.e. $\mathbf {1} _{\mathbb {Q} }(x)=1$ if $x$ is a rational number and $\mathbf {1} _{\mathbb {Q} }(x)=0$ if $x$ is not a rational number (i.e. is an irrational number).

\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 pathological function which provides counterexamples to many situations.

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