Does there exist a non-linear surjective function from a lower dimension to a high dimension?

Question

For example, for function F: R2 -> R3. Can the output space of F be R3?

If F is a linear function, the answer is obviously no. But what about non-linear functions? My intuition tells me the answer is no but is there a proof? What about continuous non-linear functions?

Answer 1

How about a space filling curve? Check out the following link. http://people.math.harvard.edu/~kupers/notes/spacefillingfunctions.pdf

Answer 2

Yes, there exist bijections between $\Bbb{R}$ and $\Bbb{R}^n$ for any $n$. So, (by composition) there are bijections between $\Bbb{R}^n$ and $\Bbb{R}^m$ for any $n,m$, so in particular for $n=2, m=3$ (and note that bijections in particular are surjections).