We have a iid sequence of random variables $X_1, X_2, \dots, X_n$, where $E(X_i) = \mu$ and $var(X_i) = \sigma^2$. The sample mean $\bar{X}$ converges to $\mu$ at rate $\sqrt{n}$ thanks to the LLN.
If we have a continuous function $f()$, the continuous mapping theorem assures that $f(\bar X)$ converges to $f(\mu)$.
My question is the following: at what rate does $f(\bar X)$ converge to $f(\mu)$?
Asymptotically I would say $\sqrt{n}$, given that $f()$ is continuous and hence locally linear. But can we have convergence rates very different from $\sqrt{n}$ in small samples?