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$