I've been reading a bit on these, not in a rigorous way yet, and it's an enjoyable read. But now I've a few questions.(adsbygoogle = window.adsbygoogle || []).push({});

As I understand it, they allow for a continuous index set in [tex]\Re[/tex] to completely cover a higher dimensional [tex]\Re[/tex][tex]^{n}[/tex]. Everywhere continuous, but nowhere differentiable, so it can't be used to represent the codomain in any way. That's kind of what non-holomorphic means, yes? That's sort of disappointing, but it also sort of makes sense.

There can be many space filling curves for the same space - infinitely many, right? I think there should be uncountably many, but I'm not sure about that, which is question 1. If so, I'd expect it could be possible for at least one such curve to be differentiable somewhere, even if only in a tiny little corner. Is this possible? That's question 2. Is there any loophole in "nowhere differentiable" that might work out as "somewhere, but not elsewhere." If the answer to question 2 is yes, which I'm not expecting, then question 3 is if that tiny section might ever possibly be smooth.

Thanks much.

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# Space filling curves: two and a half questions

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