Editing Dirichlet function (section)

== Topological properties ==
{{unordered list
| The Dirichlet function is [[nowhere continuous function|nowhere continuous]].
{{Math proof| drop=hidden|proof=*If {{mvar|y}} is rational, then {{nowrap|{{math|{{var|f}}({{var|y}}) {{=}} 1}}}}. To show the function is not continuous at {{mvar|y}}, we need to find an {{mvar|ε}} such that no matter how small we choose {{mvar|δ}}, there will be points {{mvar|z}} within {{mvar|δ}} of {{mvar|y}} such that {{math|{{var|f}}({{var|z}})}} is not within {{mvar|ε}} of {{nowrap|{{math|{{var|f}}({{var|y}}) {{=}} 1}}}}.  In fact, {{frac|1|2}} is such an {{mvar|ε}}. Because the [[irrational number]]s are [[dense set|dense]] in the reals, no matter what {{mvar|δ}} we choose we can always find an irrational {{mvar|z}} within {{mvar|δ}} of {{mvar|y}}, and {{nowrap|{{math|{{var|f}}({{var|z}}) {{=}} 0}}}} is at least {{frac|1|2}} away from 1. 
*If {{mvar|y}} is irrational, then {{nowrap|{{math|{{var|f}}({{var|y}}) {{=}} 0}}}}. Again, we can take {{nowrap|{{math|{{var|ε}} {{=}} {{frac|1|2}}}}}}, and this time, because the rational numbers are dense in the reals, we can pick {{mvar|z}} to be a rational number as close to {{mvar|y}} as is required. Again, {{nowrap|{{math|{{var|f}}({{var|z}}) {{=}} 1}}}} is more than {{frac|1|2}} away from {{nowrap|{{math|{{var|f}}({{var|y}}) {{=}} 0}}}}.}}
Its restrictions to the set of rational numbers and to the set of irrational numbers are [[constant function|constants]] and therefore continuous. The Dirichlet function is an archetypal example of the [[Blumberg theorem]].

| The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
<math display="block">\forall x \in \R, \quad \mathbf{1}_{\Q}(x) = \lim_{k \to \infty} \left(\lim_{j\to\infty}\left(\cos(k!\pi x)\right)^{2j}\right)</math>
for integer {{mvar|j}} and {{mvar|k}}. This shows that the Dirichlet function is a [[Baire function|Baire class]] 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a [[meagre set]].<ref>{{cite book
  | last = Dunham
  | first = William
  | title = The Calculus Gallery
  | publisher = [[Princeton University Press]]
  | date = 2005
  | pages = 197
  | isbn = 0-691-09565-5 }}</ref>
}}