I was asked to prove that if $f\colon A \to \mathbb{R}$ is uniformly continuous and there exists $M \in \mathbb{R}$ such that $\left | f(x) \right | \ge M$, then $g(x)= 1/f(x)$ is uniformly continuous.
But taking $A = (0,\infty)$ and $f(x)=x$ then $g(x)=1/x$ is clearly not uniformly continuous (as the derivative is not bounded).
What am I missing?
Thank you