# Is there a square-filling curve?

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.

-
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