If $f,g$ is uniformly continuous on some domain $S$,then $fg$ still uniformly continuous. I think i could find a counter example but when i try to explain why the statement fail by considering some property the uniform continuous functions have, it seems that i don't quite see(or don't quite get the big picture) why they fail. So my question is can it be explained by considering the property of the uniform continuity to explain the statement above
The definition of uniform continuity I am studying: $ \forall \epsilon>0, \exists \delta>0 s.t.\forall x,y \in S,d(x,y)<\delta \implies d(f(x),f(y))<\epsilon$
Best Answer
Well, look at the function $f(x) = x$ over the whole real line. It is uniformly continuous but its product with itself is not. Uniform continuity is about how the function changes on the domain with respect to the independent variable. If the change is too steep, like in $f(x) = x^{2}$, you are not likely to get uniform continuity. If the change is reasonably slow, like in $f(x) = x$ or $f(x) = \sqrt{x}$ for example, you are likely to get uniform continuity. Thus, while $f$ and $g$ themselves may change "reasonably" slowly, their product may change faster and hence may not be uniformly continuous.