I'm trying to prove:
If : and : are uniformly continuous functions, then is uniformly continuous.
I've gotten to the fact that
|f(x)g(x) - f(y)g(y)| <= |f(x)| * |g(x)-g(y)| + |g(y)| * |f(x) - f(y)|
But I'm not sure where to go from here...
I'm trying to prove:
If : and : are uniformly continuous functions, then is uniformly continuous.
I've gotten to the fact that
|f(x)g(x) - f(y)g(y)| <= |f(x)| * |g(x)-g(y)| + |g(y)| * |f(x) - f(y)|
But I'm not sure where to go from here...
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