Hilbert curves

Yes, I've got that part - but that only gets me as far as R * I ^ (d - 1), not R ^ d.

(d being the dimensionality of the curve.)

As I think about this there's an unfilled infinitesimal R ^ (d - 1) space orthogonal to every point on the curve.

I don't see how it is that that does not manifest as discontinuities in the filled space.

You can get from R^d to I^d (with I the open unit interval) easily by applying x -> (arctan(x) + pi/2)/pi and y -> tan(pi * y - pi/2) to each coordinate. There are also ways to go from R to the closed unit interval. You can then combine this with space-filling curves for I^d.

You can even go from R to I^d.

The problem that I'm having is in getting from R to R^d, which I still don't think is possible - look at the comment where I mention "surjections" and "bijections".

By applying the map from I^d to R^d (using the tan-arctan-construction above in each coordinate) after the map from R to R^d, you get a map from R to R^d.

I^n is just a countably infinite subset of R^n.

No, I^n is also uncountable. In fact, the usual proof for the uncountability of R is really proof of the uncountability of I, and then the natural injection into R is used to extend it to R.