TITLE: Are there spaces with non-integer Hausdorff dimension that are differentiable?
QUESTION [7 upvotes]: I realize that fractal curves usually represent non-differentiable functions in some sense, but would it to be possible to have a space that has non-integer Hausdorff dimension, and still have a reasonable notion of differentiation?  Essentially I'm curious about the ability for a space with a (pseudo)Riemannian metric, to have non-integer Hausdorff dimension.  If so, can you please provide an example and/or construction?
   If not, why?

REPLY [3 votes]: For standard differentiation on the real line, the answer is (unsurprisingly) no. The following corollary appears in the paper Sets of Fractional Dimensions (V): on Dimensional Numbers of Some Continuous Curves by A.S. Besicovitch and H.D. Ursell:
Corollary: The dimensional number of the curve $y=f(x)$, where $f(x)$ has finite derivative at all points is $1$.