Suppose the functions $f$ and $g$ are uniformly continuous on $D$. prove that $f+g$ is also uniformly continuous continuous on $D$. Show that $f⋅g$ is not necessarily uniformly continuous by a suitable example.
I proved that $f+g$ is uniformly continuous but i'm not sure I posted my attempt please read it I want to make sure that it is correct, but I need help with the example for $f⋅g$. thank you.
Your proof for $f+g$ is correct. For $fg$ take $f(x)=g(x)=x$ on $\mathbb R$.
To show that $x^{2}$ is not uniformly continuous consider the points $n, n+\frac 1 n$.