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Follow-up: Using [[Roth's theorem]] one can easily show that <math>f</math> is [[Hölder condition|Hölder continuous]] for every <math>\alpha<\tfrac12</math> on the set of irrational numbers. Using [[Hurwitz's theorem (number theory)|Hurwitz's theorem]] one can easily show that <math>f</math> is not [[Hölder condition|Hölder continuous]] for every <math>\alpha>\tfrac12</math> on the set of irrational numbers. It might be useful to add this information and the links to [[Roth's theorem]] and [[Hurwitz's theorem (number theory)|Hurwitz's theorem]].
: To be more precise, not Hölder for every <math>\alpha\ge\tfrac12</math>.
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