Suppose $X_1 , \dots, X_n$ are i.i.d. $U(\theta_1, \theta_2)$, with $\theta_1 < \theta_2$. We want to test for $$ H_0: \theta_1 \leq 0 \quad \text{versus} \quad H_1: \theta > 0. $$ It can be show that $(S, T) := (X _{(1)}, X _{(n)}) = (\min X_i, \max X_i)$ are jointly sufficient, and their joint density are $$ f _{S, T} (s, t) = \frac{n (n - 1)}{(\theta_2 - \theta_1)^n} (t - s) ^{n - 2}, \quad \theta_1 < s < t < \theta_2. $$
The book I am reading, A Graduate Course on Statistical Inference (pp. 126), then claims that the UMPU-$\alpha$ test is $$ \phi (s, t) = \begin{cases} 1 & \text{if } s > k(t) \\ 0 & \text{otherwise}, \end{cases} \qquad \alpha = \mathbb{E} _{\theta_1 = 0} [\phi (S, T) | T = t]. $$ This is derived using the Karlin-Rubin Theorem on the conditional distribution of $S | T$. Up to this part, I can still understand.
However, I am questioning on the $\color{red}{\text{continuity of the power function of } \phi}$. In order to claim that the test with nuisance parameter is UMPU, it is required that the power function $\beta _\psi (\theta) = \mathbb{E} _{\theta} [\psi (S, T)]$ is continuous in $\theta = (\theta_1, \theta_2)$ for any test function $\psi$. In this case, using the density for $(S, T)$, the power function is of the form $$ \beta _\psi (\theta) = \frac{n(n - 1)}{(\theta_2 - \theta_1)^n} \int ^{\theta_2} _{\theta_1} \int ^t _{\theta_1} \psi (s, t) (t - s) ^{n - 2} \ ds \ dt. $$ The book did not show the continuity of it, and it has been too difficult for me to prove it. Usually for the exponential family, continuity of power follows immediately. Yet, in this problem, $(\theta_1, \theta_2)$ appears both outside the integral and on the bound. I have no idea on how to proceed.
Please comment if you have any thoughts on this, or have any reference for such settings. Any help is appreciated.