I believe the Takakgi function satisfies this property. According to The Takagi Function: A Survey, the Takagi function $T$ satisfies $$T(x+h)-T(x) = O(h\log(1/|h|) \: \text{ as } \: h\to0$$ and this is the best possible estimate.

I believe the Takakgi function satisfies this property. According to The Takagi Function: A Survey, the Takagi function $T$ satisfies $$T(x+h)-T(x) = O(h\log(1/|h|) \: \text{ as } \: h\to0$$ and this is the best possible estimate.

Of course, lower bounds for Hausdorff measure are tricky but I think the result you need can be found in the paper On the Hausdorff Dimension of Some Graphs by Mauldin and Williams.