We really need some more information to solve this problem. We need some information about the set .

For example let both of these are uniformly continuous on the set let

Now consider their prouduct

on this unbounded set this is not uniformly continuous.

So if A is compact. The proof is easy! because continuous functions on compact sets are bounded.

If A is an open interval the function is uniformly continuous if and only if it can be continuously extended to the closed compact set