Continuity of log-partition function for exponential family

Question

For exponential family distributions, is the log-partition function continuous and differentiable in the whole natural parameter space?

This maybe a trivial question but I couldn't find any reference or show it rigorously. Any help is highly appreciated.

The exponential family (in canonical form) is $$f_{X}(x|\eta)=h(x)\exp \left(\eta^T T(x) - A(\eta) \right), $$ where $h(x)$, $T(x)$, and $A(\eta)$ are the base measure, sufficient statistic, and log-partition function, respectively.

In particular, the log partition function is $$ A(\eta)=\ln\left( \int_X h(x) \exp \left(\eta^T T(x) \right)dx\right)$$ and the natural parameter space is $$\mathcal{N}=\left\{\eta: \int_X h(x) \exp \left(\eta^T T(x) \right)dx < \infty \right\}.$$

Is the log-partition function $A(\eta)$ continuous for any member in the exponential family in the entire natural parameter space $\mathcal{N}$? If not, what conditions (like requiring a regular or nonsingular exponential family or only in some interior of $\mathcal{N}$) are needed for $A(\eta)$ to be continuous.

Here is what I have attempted so far to solve this problem:

1. It is straightforward to show that $A(\eta)$ is a convex function. So my first attempt was to argue from convexity to continuous. But after some reading I found that "Every function that is finite and convex on an open interval is continuous on that interval (including $\mathbb{R}^n$)".
   • By definition $A(\eta)<\infty$ for all $\eta \in \mathcal{N}$. I have both finiteness and convexity.
   • However, I couldn't find any proof that $\mathcal{N}$ is an open interval or an open subset of $\mathbb{R}^n$. Is this something trivial that doesn't need a proof? Or is it just not true?
2. A standard result for exponential family in many references is $$E[T_j(x)|\eta] = \frac{\partial A(\eta)}{\partial \eta_i}, \quad \forall i.$$ It seems it is always assumed that $A(\eta)$ is differentiable. So I thought that $A(\eta)$ must be continuous because it is differentiable. But, I wonder in what space is $A(\eta)$ differentiable. Specifically, is the space $\mathcal{N}' =\left\{\eta: |E[T(x)|\eta]| < \infty \right\}$ the same as $\mathcal{N}$ or a strict subset of the latter? If $\mathcal{N}'$ is a strict subset of $\mathcal{N}$, does it mean that $A(\eta)$ is continuous in $\mathcal{N}'$ but not necessarily in $\mathcal{N}$?

Thanks in advance for any comments and pointers.