How can an image of [0,1] fill a square? It seems to me that it would have to be DISCONTINUOUS, so it wouldn't satisfy the definition of a curve.
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The space filling curve is in itselft the limit of a process,iteraded infinitely many times. They key here is that the function built are surjective functions to the limit, which mean they fill the space.You can check out for example This article which gives an introduction to Peano's curve.