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I can prove that $A$ is closed bounded, could any one tell me $A$ is connected and dense too?thank you.

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$A$ is the closure in $\mathcal C[0,1]$ of the set $B$ where $B=\{f\in\mathcal C^1[0,1]; |f(x)|\le1\text{ and }|f'(x)|\le1\text{ for all }x\in[0,1]\}.$ Answer: closed, compact, connected, dense

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    @SimenK. edited, thank you2012-10-26

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Compactness: Arzela-Ascoli.

Connectedness: the closure of a connected set is connected (note that $B$ is convex, hence so is $A$).

Since you didn't say dense where this is impossible to answer.

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    Well, since $A$ is compact, it is quite hard for it to be dense anywhere but itself :-)2012-10-26