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A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous

Suppose that $f$:$R^m$$R^n$ sends all compact sets to compact sets,and all connected sets to connected sets.Prove that $f$ is continuous. Obviously continuous functions sends compact sets to compact,and connected to connected,but it is hard for me to prove this special opposite direction. I suppose it is really important to use that $A$ is compact in $R^n$ iff $A$ is closed and bounded but I don't know how to use it.Also,values dimensions $m$ and $n$ are arbitrary,and I guess only important thing about them is that characterisation of compact sets.

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    Thank you very much for your help.2012-12-19

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