For $f: A \subset \mathbb{R} \setminus\{0\} \to \mathbb{R} \setminus\{0\}$, is $f$ uniformly continuous on $(0, k]$ if and only if $1/f$ is uniformly continuous on $[k, +\infty)$?
The examples I've tried (e.g. $x \mapsto x^2$) suggest that it is, but I haven't proven it.
If not: Is there a weaker claim that is true? In particular, is it true for continuity (not necessarily uniform)?
Intuitively, there are many topological properties where $1/0$ behaves like $\infty$, or where $\{1/n\}$ resembles $\mathbb{N}$. It seems continuity is topological but uniform continuity is not.
This is not true. Just consider $f(x) = 1/x$ which is not uniformly continuous on $(0,k]$ but $1/f(x) = x$ definitely is.