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

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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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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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 '10 at 0:31