f(x) and g(x) are two uniformly continuous functions ,$x \in R$
if $\ lim\ xg(x) = 0 (x \rightarrow \infty)$ then i can prove that f(x)g(x) is uniformly continuous
my question is: If an $\alpha \in (0,1)$ exists,$\ lim\ |x|^{\alpha}g(x) = 0 (x \rightarrow \infty) \Longrightarrow $f(x)g(x) is uniformly continuous
if $\alpha$ doesnt exist ,then we need an example . actually I just can't find one
Let $f(x)=x$ and define a piecewise linear function $g$ as follows: $g(x)=\frac 1n$ on $(n-\frac 1{n^{2}},n+\frac 1{n^{2}})$, $g(n-\frac 2{n^{2}})=0=g(n+\frac 2 {n^{2}})$, ($g$ has a straight line graph in $[ n-\frac 2{n^{2}},n-\frac 1{n^{2}}]$ as well as $[ n+\frac 1 {n^{2}},n+\frac 2{n^{2}}]$) and $g$ is $0$ for points outside all these intervals. Then all your conditions are satisfied for every $\alpha \in (0,1)$ and $h(x)\equiv f(x)g(x)$ is not uniformly continuous: $h(n-\frac 1 {n^{2}})=1-\frac 1 n$ and $h(n-\frac 2 {n^{2}})=0$.