I have the function f: A-R such that f is uniformly continuous. abs(f(x)) greater than or = to k greater than 0 for all x in A. How would I show that (1/f) is also uniformly continuous on A?

I also need to find an example where the conclusion fails for a function which only satisfies abs(g(x)) greater than 0 for all x in A. Would the function

g(x)=(x-1)/(x+1) where 1/g is (x+1)/(x-1) be an example since it would not be continuous at 1?