4
$\begingroup$

Possible Duplicate:
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.

  • 3
    an answer to your question is here http://math.stackexchange.com/questions/220410/a-characterization-of-real-valued-functions-on-bbb-r-which-are-contiunuous2012-12-18
  • 1
    I edited that question to generalize it to reflect the existing generalized answers, so this question can now be closed as a duplicate of that one.2012-12-18
  • 0
    Thank you very much for your help.2012-12-19

0 Answers 0