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This question is inspired by How kinky can a Jordan curve get?

What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is the least upper bound attained by some function?

It may be noted that the area (2-dimensional Hausdorff measure) of a function graph is zero. However, this does not rule out the possible existence of a function graph of dimension two.

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  • $\begingroup$ >>It may be noted that the area (2-dimensional Hausdorff measure) of a function graph is zero. Really? I don't know how to prove it if the function is non-measurable. $\endgroup$
    – user69664
    Commented Mar 25, 2015 at 0:07
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    $\begingroup$ @Catcat I had continuous functions in mind, as indicated in the second paragraph. $\endgroup$
    – Harald Hanche-Olsen
    Commented Mar 25, 2015 at 12:59

3 Answers 3

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The answer is 2.

Besicovitch and Ursell, Sets of fractional dimensions (V): On dimensional numbers of some continuous curves. J. London Math. Soc. 12 (1937) 18–25. doi:10.1112/jlms/s1-12.45.18

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Here's is one example with Hausdorff dimension $2$.

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    $\begingroup$ Thank you. Have you checked the proof yourself? I ask because that journal (Chaos, Solitons and Fractals) has a somewhat dubious reputation (see Physics World, january 2009 and Nature, November 2008; also zeit.de/2009/03/N-El-Naschie?page=1, arstechnica.com/science/news/2008/11/…), scienceblogs.com/pontiff/2008/11/…) $\endgroup$
    – Harald Hanche-Olsen
    Commented May 8, 2010 at 21:45
  • $\begingroup$ To follow up on my own comment, at least the paper looks serious. So I think it's probably okay. $\endgroup$
    – Harald Hanche-Olsen
    Commented May 8, 2010 at 21:50
  • $\begingroup$ I haven't checked it carefully. Thanks for the articles, I had never heard of this before. $\endgroup$
    – Gjergji Zaimi
    Commented May 8, 2010 at 22:16
  • $\begingroup$ 2 is the right answer, but of course 2007 is not the earliest example $\endgroup$
    – Gerald Edgar
    Commented May 8, 2010 at 23:58
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    $\begingroup$ Once you get arbitrarily close to 2, just use disjoint intervals and put graphs on them with dimensions $\gt 2-1/n$ $\endgroup$
    – Gerald Edgar
    Commented May 9, 2010 at 0:31
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It seems that most of the continuous functions over the unit interval have a graph of Hausdorff dimension 2. To be more precise, it is proved in https://projecteuclid.org/euclid.rae/1366030629 (The Hausdorff Dimension of Graphs of Prevalent Continuous Functions by Jonathan M. Fraser and James T. Hyde) that the set of continuous function whose graph has Hausdorff dimension 2 is prevalent.

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  • $\begingroup$ For the earliest published results (that I know of) giving the various fractal dimensions of graphs of most (Baire category, sup norm) continuous functions, see the references in my diagram in this paper. $\endgroup$
    – Dave L Renfro
    Commented Mar 27, 2018 at 14:51

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