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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.

kh7
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    So the map $f \colon \mathbb{R} \to \mathbb{R}^2$ which takes $t$ to $(t,0)$ should not be continuous? – Matthew Leingang May 01 '19 at 09:18
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    I think you want your map to genueinly higher dimensional range but you did not mention that. See https://en.wikipedia.org/wiki/Space-filling_curve for such functions. – Kavi Rama Murthy May 01 '19 at 09:20
  • @MatthewLeingang well...I guess it is time for bed if I can't see that haha. Thank you. – kh7 May 01 '19 at 09:27

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