OK, I think I've solved the first one:

Fix

. Now choose

such that

implies

. (Possible since

uniformly.) Next, choose

such that

implies

. (Possible since

is continuous at

.) Now, choose

such that

implies

. (Possible since

.) Let

be the maximum of

. Then

implies

which again implies

, but it also implies

. But then

, so

.

After some consideration it seems to me that the second problem is a direct consequence of the first, because if we assume that

does converge uniformly to

we get a contradiction. Is this correct?