Suppose that $X$ is a random vector and $x_0$ is a fixed vector such that $$ X-x_0=O_p(n^{-1/2}).\tag{$*$} $$ Let $Y=g(X)$ where $g$ has a continuous gradient that is nonzero at $x_0$. Let $y_0=g(x_0)$. My reading claims $\boxed{Y-y_0=O_p(n^{-1/2})}$. Why is this the case? (And if this is a standard result, can you please point a source where I can look it up?)
I'm trying to go about this via first principles. ($*$) implies that $$ \forall\epsilon>0\implies\exists M>0:\Pr(\sqrt{n}|X-x_0|>M)<\epsilon\;\forall n\geq 1.\tag{$**$} $$ Ideally, I would construct $N$, depending $\epsilon$, $M$, and $g$ somehow so that I would have $$ \forall\epsilon>0\implies\exists N>0:\Pr(\sqrt{n}|Y-y_0|>N)<\epsilon\;\forall n\geq 1.\tag{$***$} $$ My issue is that continuity and ($**$) imply there is $N$ for each $n$ but I don't have an $N$ that works for all $n$.