Formal proof that hilbert curve is space-filling

Ryodansm

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Mar 3, 2021
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Hello,
I am searching for a formal proof (or some clues) that the well-defined continuous (supposed already demonstrated) Hilbert curve is space-filling (which means that every point of the unit square is an output by the limit function ?
I hope someone can provide a link to a formal proof or at least point me in the right direction.
Thanks in advance.
 
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