Just a simple question: Can a function mapping $\mathbb{R}^m$ into $\mathbb{R}^n$ be a continuous function when m < n?
My gut says "No." My brain says "Go to bed already."
I'm trying to prove something using a result that holds for continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$, but it seems like that won't be an appropriate assertion if I have to worry about continuous functions that map $\mathbb{R}$ to a higher dimension.