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...

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- Apr 3rd 2011, 05:45 AMmosesMultiplication of uniformly continuous functions
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... - Apr 3rd 2011, 08:02 AMTheEmptySet
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