Dirichlet Function
Let 
and 
be real
numbers (usually taken as 
and 
). The Dirichlet
function is defined by
 |
(1)
|
and is discontinuous everywhere. The Dirichlet function can be written analytically as
 |
(2)
|
Because the Dirichlet function cannot be plotted without producing a solid blend of lines, a modified version, sometimes itself known as the Dirichlet function (Bruckner et al. 2008), Thomae function (Beanland et al. 2009), or small Riemann function (Ballone 2010, p. 11), can be defined as
 |
(3)
|
(Dixon 1991), illustrated above. This function is continuous at irrational 
and discontinuous
at rational 
(although a small interval around an
irrational point 
contains infinitely many rational points,
these rationals will have very large denominators). When viewed from a corner along
the line 
in normal perspective, a quadrant
of Euclid's orchard turns into the modified Dirichlet
function (Gosper).
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