Let $f: \mathbb R^n \rightarrow \mathbb R^n$, where $n> 1$ be a bijective map such that every line is mapped to a line.

Is $f$ continuous?

I think it is, but the proof isn't immediately obvious to me.
Related to <a href="http://math.stackexchange.com/q/11232/3638">this question on math.SE</a>.

Feel free to retag.