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*: Thus for all <math>n</math>, <math>\left| \frac{f(b_n)-f(x_0)}{b_n-x_0} \right| \ge \frac{1/n - 0}{1/n} = 1</math>. Therefore we obtain <math>\liminf_{n\to\infty} \left| \frac{f(b_n)-f(x_0)}{b_n-x_0} \right| \ge 1 \ne 0</math> and so <math>f</math> is not differentiable at any irrational number <math>x_0</math>.
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
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