Let $\{X_i\}$ be iid random variables on $\mathbb R$ $X_i\sim X$, let $f: \mathbb R\to \mathbb R$ be a continuous function s.t. expectations $\mathrm EX$ and $\mathrm Ef(X)$ are finite. Does it follow that $$ \lim_{n\to\infty}\mathrm Ef\left(\frac{X_1+\ldots+X_n}n\right)=f(\mathrm EX)? $$
If not, what are the condition for this to hold? If random variable $X$ is bounded a.s. it seems to follow from Hoeffding's inequality.