Proving expectation and variance of a function of a random variable tends to a fix point

Question

Given $f:\mathcal{X} \rightarrow \mathbb{R}$ is a continuous function and $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ ($x^\star$ is a fix number), $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$. How can we prove that \begin{align} \mathbb{E}_{Q(X)}[f(X)] \rightarrow f(x^\star), \quad \text{and}\qquad \mathbb{V}\text{ar}_{Q(X)}[f(X)] \rightarrow 0? \end{align}