Definition (Regular Model [1, p. 203]). A standard statistical model $\big( X, \mathcal F, (\mathbb P_{\vartheta})_{\vartheta \in \Theta}\big)$, where $\Theta \subset \mathbb R$ is an open interval, $X \ne \emptyset$, $\mathcal F$ is a $\sigma$-algebra on $ X$ and $\mathbb P_{\vartheta}$ is a probability measure on $( X, \mathcal F)$ for every $\vartheta \in \Theta$ such that there exists a measure $\mu_0$ with $\mathbb P_{\vartheta} \ll \mu_0$ (absolute continuity) for all $\vartheta \in \Theta$ is regular, if the likelihood function $$\rho \colon X \times \Theta \to \mathbb{R}, \qquad (x, \vartheta) \mapsto \frac{\text{d} \mathbb P_{\vartheta}}{\text{d} \mu_0}(x)$$ (Radon-Nikodym derivative) is positive and continuously differentiable (almost everyhwere) with respect to $\vartheta$.
Then the score $$ U_{\vartheta}(x) := \frac{\partial}{\partial \vartheta} \ln\big(\rho(x, \vartheta)\big) = \frac{\frac{\partial}{\partial \vartheta} \rho(x, \vartheta)}{ \rho(x, \vartheta)} $$ is well defined.
For a regular model, we also require that the Fisher information of the model $I(\vartheta) := \mathbb{V}_{\vartheta}[U_{\vartheta}]$, where $V_{\vartheta}$ is the variance with respect to $\mathbb P_{\vartheta}$, is always an element of $(0, \infty)$ and that \begin{equation} \int_{ X} \frac{\partial}{\partial \vartheta} \rho(x, \vartheta) \, \text{d}{\mu_0(x)} \overset{!}{=} \frac{\partial}{\partial \vartheta} \underbrace{\int_{ X} \rho(x, \vartheta) \, \text{d}{\mu_0(x)}}_{= 1} = 0 \end{equation} holds, where we require that the left side is well defined, that is, that $\frac{\partial}{\partial \vartheta} \rho(\cdot, \vartheta)$ is integrable with respect to $\mu_0$.
It is not difficult to show that $\mathbb{E}_{\vartheta}[U_{\vartheta}^2] = 0$ and thus that $I$ is lower continuous due to Fatou's Lemma).
My question: Is $I$ even continuous?
Remark. The proof of the Cramér-Rao inequality one shows that if that the inequality is an equality, then $I$ is continuous, but the requirements of that theorem are, among others, that the estimator is regular and that the expected value of the estimator with respect to $\mathbb P_{\vartheta}$ is continuously differentiable with respect to $\vartheta$ with pointwise non-vanishing derivative.
1 Georgii, Hans-Otto, Stochastics. Introduction to probability and statistics. Translated by Marcel Ortgiese, Ellen Baake and the author., de Gruyter Textbook. Berlin: de Gruyter (ISBN 978-3-11-019145-5/pbk; 978-3-11-020676-0/ebook). ix, 370 p. (2008). ZBL1270.62005.