Prove that if , is not compact. Then there exists a continuous function which isn't bounded on

Any help would be appreciated.

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- May 3rd 2009, 12:16 PMskamoniCompact continuity question.
Prove that if , is not compact. Then there exists a continuous function which isn't bounded on

Any help would be appreciated. - May 3rd 2009, 01:21 PMOpalg
- May 3rd 2009, 01:27 PMThePerfectHacker
Two possibilities: (i) either F is not closed, (ii) F is not bounded.

If F is not closed then there exists so that . Now define .

This function is continous and unbounded since gets arbitrary close to .

If F is not bounded then for some define .

Certainly this is not bounded since gets arbitrary away from .

EDIT:**Opalg**beat me too it :mad:.

By the way I think it is possible to find a function :eek:, do you agree ?