Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem:
Definition: The $p$-Wasserstein metric $W_{p}(\mu,\nu)$ between $\mu,\nu\in\mathcal{P}_{p}(\Xi)$ is defined by $$W_{p}^{p}(\mu,\nu):=\min_{\Pi\in\mathcal{P}(\Xi\times\Xi)}\left\{\int_{\Xi\times\Xi}d^{p}(\xi,\zeta)\Pi(d\xi,d\zeta)\: :\: \Pi(\cdot \times\Xi)=\mu(\cdot),\: \Pi(\Xi\times\cdot)=\nu(\cdot)\right\}$$ where $$\mathcal{P}_{p}(\Xi):=\left\{\mu\in\mathcal{P}(\Xi)\: :\: \int_{\Xi}d^{p}(\xi,\zeta_{0})\mu(d\xi) < \infty\ \mbox{for some }\zeta_{0}\in\Xi\right\}$$ where $d$ is a metric on $\Xi$.
The question: Given a function $f:\Xi\rightarrow \mathbb{R}$ we consider the function $F:\mathcal{P}_{p}(\Xi)\rightarrow \mathbb{R}$ defined by $F(\nu):=\int_{\Xi}f(\xi)\nu(d\xi)$. Is $F$ continuous with respect to the metric $W_{p}$? That is, given a sequence $\{\nu_{n}\}_{n=1}^{\infty}\subset \mathcal{P}_{p}(\Xi)$ and $\nu\in\mathcal{P}_{p}(\Xi)$ such that $W_{p}(\nu_{n},\nu)\rightarrow 0$, then do we have $\left|F(\nu_{n})-F(\nu)\right|\rightarrow 0 $?
Remark: If $f$ is bounded then the answer to the previous question is affirmative, this is a consequence of Definition 6.8 and Theorem 6.9 of Cedric Villani's book. The problem is when $f$ is not bounded, for example $f(\xi)=\xi^{2}-\xi$ with $\Xi=\mathbb{R}$.