The Dirichlet Function is not continuous at any point on the real line.
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Proof:
Left as Exercise. Show that g can not be continuous at a rational
number by considering sequences of irrational numbers. Then show
that g can not be continuous at an irrational number by considering
sequences of rational numbers.
Interactive Real Analysis, ver. 1.9.5 (c) 1994-2007, Bert G. Wachsmuth Page last modified: Mar 28, 2007