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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.

flag	asked May 8 2010 at 21:03 Harald Hanche-Olsen 5,750●1●13●34

Harald Hanche-Olsen · Accepted Answer · 2010-05-09 14:04:03Z UTC

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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

edited May 9 2010 at 14:04

Harald Hanche-Olsen

answered May 9 2010 at 0:30

Gerald Edgar
15.2k●1●28●67

Gjergji Zaimi · Answer 2 · 2010-05-08 21:24:33Z UTC

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Here is one example with Hausdorff dimension 2.

answered May 8 2010 at 21:24

Gjergji Zaimi
37.5k●2●72●149

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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/…) – Harald Hanche-Olsen May 8 2010 at 21:45

To follow up on my own comment, at least the paper looks serious. So I think it's probably okay. – Harald Hanche-Olsen May 8 2010 at 21:50

I haven't checked it carefully. Thanks for the articles, I had never heard of this before. – Gjergji Zaimi May 8 2010 at 22:16

2 is the right answer, but of course 2007 is not the earliest example – Gerald Edgar May 8 2010 at 23:58

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Once you get arbitrarily close to 2, just use disjoint intervals and put graphs on them with dimensions $\gt 2-1/n$ – Gerald Edgar May 9 2010 at 0:31

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How big can the Hausdorff dimension of a function graph get?

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