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If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.

I would like to see an example of such a function with a NON $\sigma$ finite graph (with respect to $\mathscr{H}^{1}$).

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    $\begingroup$ Do you know such examples exist or are you implicitly asking whether or not they do? $\endgroup$
    – Thompson
    Commented Jun 5, 2017 at 23:45
  • $\begingroup$ I do not know if such example exists. However I would be surprised if it does not, because in several papers I have seen non trivial proofs for the $\sigma$ finiteness of the graphs of functions in special subclasses of the class $\bigcap_{0 < \alpha < 1}\mathscr{C}^{0,\alpha}$, everyone using some special feature of the subclass. $\endgroup$
    – Longyearbyen
    Commented Jun 6, 2017 at 8:04
  • $\begingroup$ Did you find the answer? I am sure there is an example with a non $\sigma$-finite 1-measure. $\endgroup$
    – Piotr Hajlasz
    Commented May 17, 2020 at 18:34

1 Answer 1

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I believe the Takakgi function satisfies this property. According to The Takagi Function: A Survey, the Takagi function $T$ satisfies $$T(x+h)-T(x) = O(h\log(1/|h|) \: \text{ as } \: h\to0$$ and this is the best possible estimate.

Of course, lower bounds for Hausdorff measure are tricky but I think the result you need can be found in the paper On the Hausdorff Dimension of Some Graphs by Mauldin and Williams.

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  • $\begingroup$ Thank you for your answer. In this survey they claim that the graph is $\sigma$ finite, see p.16. Actually the fact that the takagi function has $\sigma$ finite graph was the initial motivation for my question. $\endgroup$
    – Longyearbyen
    Commented Jun 5, 2017 at 22:37

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