# Thread: Multiplication of uniformly continuous functions

1. ## Multiplication of uniformly continuous functions

I'm trying to prove:

If $f$: $A --> R$ and $g$: $A --> R$ are uniformly continuous functions, then $f * g$ 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...

2. Originally Posted by moses
I'm trying to prove:

If $f$: $A --> R$ and $g$: $A --> R$ are uniformly continuous functions, then $f * g$ 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 $A$.

For example let $f(x)=g(x)=x$ both of these are uniformly continuous on the set $[0,\infty)$ let $\delta =\epsilon$

Now consider their prouduct $h:[0,\infty) \to \mathbb{R},h(x)=f(x)g(x)=x^2$

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 $(a,b)$ the function is uniformly continuous if and only if it can be continuously extended to the closed compact set $[a,b]$