my professor gave me this problem that I can't solve at all.. is there anyone who can help me?
Assume that $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is a (globally) Lipschitz continous function
i.e. $\exists L$ : $\forall x,y$ $\epsilon\mathbb{R}$ $|\phi(x)-\phi(y)|\leq L|x-y|$
1) Demonstrate that if $f_{n}$ is a succession of continous functions that converges uniformly to $f:\mathbb{R}\rightarrow\mathbb{R}$, so
$\phi\circ f_{n}$ converges uniformly to $\phi\circ f$
2) What can we say about the succession $\phi\circ f$ , is we assume that $\phi$ is a lipschitz function but just locally and $f$ is bounded?
3) What can we say about the succession $\phi\circ f$ , if we only assume that $\phi$ is a continous function?
Let me know if there's someone who can help me! thanks a lot