The continuous mapping theorem states that if $X_{n} \to X$ in probability and $g$ is continuous almost surely then $g(X_{n}) \to g(X)$ in probability.
In little-O notation this is $$X_{n} - X = o_{P}(1) \Rightarrow g(X_{n}) - g(X) = o_{P}(1).$$
Are there any extensions for bounded in probability? More specifically,
If $X_{n} - X = O_{P}(a_{n})$, what are conditions on $g$ which give $g(X_{n}) - g(X) = O_{P}(a_{n})$?
If I have a sequence of functions $g_{n}$ and $X_{n} - X = O_{P}(a_{n})$, what are minimal conditions on $g_{n}$ to give $g_{n}(X_{n}) - g_{n}(X) = O_{P}(a_{n})$?
This is for self study so textbook references where I can read further are appreciated.