TITLE: How big can the Hausdorff dimension of a function graph get?
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: 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.