A tiny change in your coordinate will mean a tiny change in your physical position, but a tiny change in your physical position will almost always mean a quite large (and basically not predictable without doing an involved calculation) change in your coordinate.

In the purely mathematical sense, this means something like "continuous except at points with at least one rational coordinate".

In the programming sense, where everything must be approximated and you're mapping a 2-d lattice to a 1-d lattice, then it's a bit tricky to formulate, but something like: if you take the maximum distance in 1-d space between a 2-d point and its eight "neighbors", then the probability of this distance being > N should be O(1/sqrt(N)). Concretely,

P(point has an adjacent point whose Hilbert coordinate disagrees past the highest n bits) == P(d > 4^n) < 4*(1/2)^n

Here is the plot: https://gist.github.com/justinvanwinkle/22a15bb33536095b05dc...

Notice the Log y axis. I don't you think you could consider this even roughly continuous. This is only even bounded because I only took points within the 256x256 grid, if I let it run over say a 1000x1000 grid the hilbert distances wouldn't plot well even on a log plot.

The second plot I took any random point within euclidian distance 8 from a random first point, and didn't restrict it to stay in the same 256x256 grid. 10k points total. Even at distance 1, you get hilbert distances like:

4469269309980865699858008332735282042493121940197464459241187147907254676691182325600624766055634475896677286062162016951251294270648856648811216335320387

I'm confused about the second plot; what order is the curve? It seems weird that there are no points between 10^12 and 10^156.

The difference would be more pronounced on a non-log scale, except the huge numbers at the top of the range would squish the graph down so you couldn't see what was going on.

The first graph isn't just not continuous, it's not even 1-1. (I'm ignoring the fact that is also discrete so that any formal sense of 'continuity' can't exist. Even if you tried to make it continuous by filling in the gaps it would be impossible.)