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Let $f: \mathbb R^n \rightarrow \mathbb R^n$, where $n> 1$ be a bijective map such that the image of 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 this question on math.SE.

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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?