This question is inspired by [How kinky can a Jordan curve get?](https://mathoverflow.net/questions/23954/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.



How big can the Hausdorff dimension of a function graph get?
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](http://dx.doi.org/10.1112/jlms/s1-12.45.18)


