Quotients of uniformly continuous functions Suppose $A \subset \mathbb{R}$ , $f,g : A \to \mathbb{R} $ be two uniformly continuous functions on $A$ with $g(x) \neq 0 $. We know that $f(x)/ g(x) $ is not necessarily uniformly continuous on $A$. For example, $f(x) = 1 $, $g(x) = x $ and $A = (0,1) $ works wince $1/x$ is not uniformly continuous $(0,1)$ So, my question is: under what condition on $A$ is $f(x)/g(x) $ uniform continuous on $A$ necessarily ? I think it is when $A$ is compact, how can we prove that quotient indeed works for compact sets?

If $A$ is compact then $\frac fg$ will be uniformly continuous because as long as $g \
ot= 0$ on $A$, $\frac 1g$ will be continuous, and $\frac fg$ is then the product of two continuous functions on a compact set, hence uniformly continuous.

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$A$ merely closed isn't enough. For instance, $f(x) = x$ and $g(x) = \frac{1}{x}$ are both uniformly continuous on $[1,\infty)$, but $\frac fg(x) = x^2$ is not.