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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 http://en.wikipedia.org/wiki/Space_filling_curve for an introduction, and feel free to ask back with questions about what you read.2012-09-22

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