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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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It may seem that it would have to be discontinuous, but in fact it doesn't. There are continuous functions $f : [0,1] \to [0,1]^2$ which are surjective. See en.wikipedia.org/wiki/Space_filling_curve for an introduction, and feel free to ask back with questions about what you read. –  Nate Eldredge Sep 22 '12 at 15:57
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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.

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