Editing Dirichlet function (section)

{{Short description|Indicator function of rational numbers}}
In [[mathematics]], the '''Dirichlet function'''<ref>{{springer|title=Dirichlet-function|id=p/d032860}}</ref><ref>[http://mathworld.wolfram.com/DirichletFunction.html Dirichlet Function — from MathWorld]</ref> is the [[indicator function]] <math>\mathbf{1}_\Q</math> of the set of [[rational number|rational numbers]] <math>\Q</math>, i.e. <math>\mathbf{1}_\Q(x) = 1</math> if {{mvar|x}} is a rational number and <math>\mathbf{1}_\Q(x) = 0</math> if {{mvar|x}} is not a rational number (i.e. is an [[irrational number]]).
<math display="block">\mathbf 1_\Q(x) = \begin{cases}
1 & x \in \Q \\
0 & x \notin \Q
\end{cases}</math>

It is named after the mathematician [[Peter Gustav Lejeune Dirichlet]].<ref>{{cite journal| first = Peter Gustav | last = Lejeune Dirichlet | title = Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données| journal = Journal für die reine und angewandte Mathematik |volume = 4 | year = 1829 | url = https://eudml.org/doc/183134 | pages = 157–169}}</ref> It is an example of a [[Pathological (mathematics)|pathological function]] which provides counterexamples to many situations.