Need help with uniform continuity

Question

I came across this problem which I can't solve, suppose we have $f(x) = x\sqrt{(1+x)/(1-x)}$ , is it uniform continuous in $[-1, 1)$, $[-1, 0)$, and $[0, 1)$ , what I've seen in most places that does the trick is somehow starting from $|f(x_1)-f(x_2)|$ to get something that has $|x_1-x_2|$ in it and it all works out from there on but I can't seem to pull it off here. Thanks in advance for your help.

Answer 1

The mathmetical definition of a function being uniformly continuos is that for every $\mu$ > 0, there exists $\beta$ > 0, such that for every x,x' belonging to $I$, if |x-x'|<$\beta$ $\implies$ |$f$(x)-$f$(x')|< $\mu$! You shall search for $\beta$ satisfying the implication! $I$ being the domain!