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==Dirichlet function==
{{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''', named after German mathematician [[Peter Gustav Lejeune Dirichlet]].<ref>{{cite journal |first=Lejeune Dirichlet |last=P.G. |title=Sur la convergence des séries trigonométriques qui servent à répresenter une fonction arbitraire entre des limites donées. [On the convergence of trigonometric series which serve to represent an arbitrary function between given limits] |journal=Journal für die reine und angewandte Mathematik |volume=4 |pages=157–169 |year=1829 |url=https://eudml.org/doc/183134}}</ref> This function is denoted as ''I''<sub>'''Q'''</sub> or ''1''<sub>'''Q'''</sub> and has [[___domain of a function|___domain]] and [[codomain]] both equal to the [[real number]]s. ''I''<sub>'''Q'''</sub>(''x'') equals 1 if ''x'' is a [[rational number]] and 0 if ''x'' is not rational. If we look at this function in the vicinity of some number ''y'', there are two cases: ▼*If ''y'' is rational, then {{nowrap|1=''f''(''y'') = 1}}. To show the function is not continuous at ''y'', we need to find an ''ε'' such that no matter how small we choose ''δ'', there will be points ''z'' within ''δ'' of ''y'' such that ''f''(''z'') is not within ''ε'' of {{nowrap|1=''f''(''y'') = 1}}. In fact, 1/2 is such an ''ε''. Because the [[irrational number]]s are [[dense set|dense]] in the reals, no matter what ''δ'' we choose we can always find an irrational ''z'' within ''δ'' of ''y'', and {{nowrap|1=''f''(''z'') = 0}} is at least 1/2 away from 1.
*If ''y'' is irrational, then {{nowrap|1=''f''(''y'') = 0}}. Again, we can take {{nowrap|1=''ε'' = 1/2}}, and this time, because the rational numbers are dense in the reals, we can pick ''z'' to be a rational number as close to ''y'' as is required. Again, {{nowrap|1=''f''(''z'') = 1}} is more than 1/2 away from {{nowrap|1=''f''(''y'') = 0}}.
In less rigorous terms, between any two irrationals, there is a rational, and vice versa.
▲One example of such a function is the [[indicator function]] of the [[rational number]]s, also known as the '''Dirichlet function''', named after German mathematician [[ Peter Gustav Lejeune Dirichlet ]].<ref>{{cite journal |first=Lejeune Dirichlet |last=P.G. |title=Sur la convergence des séries trigonométriques qui servent à répresenter une fonction arbitraire entre des limites donées. [On the convergence of trigonometric series which serve to represent an arbitrary function between given limits] |journal=Journal für die reine und angewandte Mathematik |volume=4 |pages=157–169 |year=1829 |url=https://eudml]. org/doc/183134}}</ref> This function is denoted as ''I''<sub>'''Q'''</sub> or ''1''<sub>'''Q'''</sub> and has [[___domain of a function|___domain]] and [[codomain]] both equal to the [[real number]]s. ''I''<sub>'''Q'''</sub>(''x'') equals 1 if ''x'' is a [[rational number]] and 0 if ''x'' is not rational. If we look at this function in the vicinity of some number ''y'', there are two cases:
The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
InMore generalgenerally, if ''E'' is any subset of a [[topological space]] ''X'' such that both ''E'' and the complement of ''E'' are dense in ''X'', then the real-valued function which takes the value 1 on ''E'' and 0 on the complement of ''E'' 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>▼:<math>f(x)=\lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)\right)^{2j}\right)</math>
for integer ''j'' and ''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>
▲In general, if ''E'' is any subset of a [[topological space]] ''X'' such that both ''E'' and the complement of ''E'' are dense in ''X'', then the real-valued function which takes the value 1 on ''E'' and 0 on the complement of ''E'' will be nowhere continuous. Functions of this type were originally investigated by [[Peter Gustav Lejeune Dirichlet]].
==Hyperreal characterisation==
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