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