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{{main article|Dirichlet function}}
One example of such a function is the [[indicator function]] of the [[rational number]]s, also known as the [[Dirichlet function]]. This function is denoted as <math>\mathbf{1}_\Q</math> and has [[___domain of a function|___domain]] and [[codomain]] both equal to the [[real number]]s. By definition, <math>\mathbf{1}_\Q(x)</math> is equal to <math>1</math> if <math>x</math> is a [[rational number]] and it is <math>0
More generally, if <math>E</math> is any subset of a [[topological space]] <math>X</math> such that both <math>E</math> and the complement of <math>E</math> are dense in <math>X,</math> then the real-valued function which takes the value <math>1</math> on <math>E</math> and <math>0</math> on the complement of <math>E</math> will be nowhere continuous. Functions of this type were originally investigated by [[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>
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===Other functions===
==Hyperreal characterisation==
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