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Does there exist a continuous function $F$ on $[0,1]$ which is not Hölder continuous of order $\alpha$ at any point $X_{0}$ on $[0,1]$. $0 < \alpha \le 1$.

I am trying to prove that such a function does exist. also I couldn't find a good example.

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    Such functions certainly exist. In fact, *almost all* continuous functions in the Baire category sense (using sup norm on the space of continuous functions) fail to have a positive pointwise Holder condition at each point, a result that was proved by Auerbach and Banach in 1931 [*Uber die Höldersche Bedingung*, Studia Mathematica 3, pp. 180-184]. I don't know of a "formula" for such a function off-hand, however.2012-03-28
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    @AlexJ. There are several examples on the wikipedia page.2012-03-28

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