Prove that if $\displaystyle F\subset\mathbb{R}$, is not compact. Then there exists a continuous function $\displaystyle f:F\rightarrow\mathbb{R}$ which isn't bounded on $\displaystyle F$

Any help would be appreciated.

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- May 3rd 2009, 11:16 AMskamoniCompact continuity question.
Prove that if $\displaystyle F\subset\mathbb{R}$, is not compact. Then there exists a continuous function $\displaystyle f:F\rightarrow\mathbb{R}$ which isn't bounded on $\displaystyle F$

Any help would be appreciated. - May 3rd 2009, 12:21 PMOpalg
- May 3rd 2009, 12:27 PMThePerfectHacker
Two possibilities: (i) either F is not closed, (ii) F is not bounded.

If F is not closed then there exists $\displaystyle p\in \partial F$ so that $\displaystyle p\not \in F$. Now define $\displaystyle f(x) = \tfrac{1}{|p-x|}$.

This function is continous and unbounded since $\displaystyle x$ gets arbitrary close to $\displaystyle p$.

If F is not bounded then for some $\displaystyle p\in F$ define $\displaystyle f(x) = |p-x|$.

Certainly this is not bounded since $\displaystyle x$ gets arbitrary away from $\displaystyle p$.

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

By the way I think it is possible to find a $\displaystyle \mathcal{C}^{\infty}$ function :eek:, do you agree ?