Let $ f:\mathbb R^n \to \mathbb R^m$. If $f$ preserves connectedness and compactness then $f$ is continuous. How can this be proven? I don't really know where to start.
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You have to prove that the image sets of connected and compact subsets of $R^n$ are connected and compact sets in $R^m$ respectively. – Mathemagician1234 Mar 29 '15 at 23:15
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@Mathemagician1234 I think he needs to assume those things and show $f$ is continuous. – Gregory Grant Mar 29 '15 at 23:17
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@GregoryGrant what you said. – Petar Markovic Mar 29 '15 at 23:18
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The answer is here: http://math.stackexchange.com/questions/220410/a-characterization-of-functions-from-mathbb-rn-to-mathbb-rm-which-are-co – shalin Mar 30 '15 at 01:30