Continuity and differentiability of the cumulant-generating function

Question

Let $X_1, X_2, \dots$ be i.i.d. random variables with $E[X_i] = \mu$. Define the moment-generating function by $\varphi(\theta) = E[\exp(\theta X_i)]$ for $\theta \geq 0$. Also, let $\theta_+ = \sup\{\theta: \varphi(\theta) < \infty\}$ and $\theta_- = \inf\{\theta: \varphi(\theta) < \infty\}$, so that $\varphi(\theta) < \infty$ for $\theta \in (\theta_-, \theta_+)$. Let $\kappa(\theta) = \log \varphi(\theta)$, the cumulant-generating function.

Now, my goal is to show that $\kappa$ is continuous at $0$ and differentiable on $(0,\theta_+)$. The steps are as follows (from Lemma 2.7.2 in Durrett, Probability: Theory and Examples):

However, several of the steps outlined there are confusing to me.

1. While proving (i), why do we need to assume $0 < \theta < \theta_0 < \theta_-$? Wouldn't that render $1+e^{\theta_0 x}$ nonintegrable (since $\theta_0 < \theta_-$), so that the dominated convergence theorem is inapplicable here?
2. If $e^{\theta_0 x}$ is indeed integrable, then why do we need $1 + e^{\theta_0 x}$ to be the dominating function instead of simply $e^{\theta_0 x}$? Doesn't it follow from the monotonicity of the exponential function that $e^{\theta x} \leq e^{\theta_0 x}$, since $\theta < \theta_0$?
3. For (ii), how do you show that $|hx|e^{h_0}x$ is integrable (with respect to the distribution function $F$)?
4. What is the appropriate dominating function for $xe^{\theta x}$ to prove (iii)? Is the text suggesting that we use $1+e^{\theta_0 x}$ again for that? But the graph of $xe^{\theta x}$ and $1+e^{\theta_0 x}$ doesn't seem to show the latter as a dominating function (or is it because we only care for $\theta$ very close to $0$?). For another candidate function, I'm thinking of $e^{(1+\theta_0) x}$ for $0 < \theta < \theta_0 < \theta_+$. But again, I am unsure on how to prove that it is indeed integrable.

After clearing the doubts on those points, I suppose it's easy to establish (iii); since $\lim\limits_{\theta \to 0} \varphi(\theta) = 1$ and $\lim\limits_{\theta \to 0} \varphi'(\theta) = E[X_i] = \mu$, we have $\kappa'(\theta) = \frac{\varphi'(\theta)}{\varphi(\theta)} \to \mu$ as $\theta \to 0$. So, in general, my main difficulty lies in determining the integrability of functions when integrating with respect to a distribution function. Any help on the aforementioned questions is greatly appreciated.

Edit:

• Now I think that the assumption $\theta_0 < \theta_-$ in (i) is a mistake. Would it be fine if I replace it with some $\theta_0 \in (\theta_-, \theta_+)$ such that $\varphi(\theta_0) = E[e^{\theta_0 X_i}] < \infty$? That way, for $0 < \theta < \theta_0$ we have $e^{\theta x} \leq 1+e^{\theta_0 x}$, so $e^{\theta_0 X_i}$ is an integrable dominating function for $e^{\theta X_i}$ if $\theta$ is small enough.