Is the following statement true? If so, could anyone provide a reference?
Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, $|f(x, \alpha)| \leq g(x)$ for all $x \in (a, b)$, and the improper integral $\int_a^b g(x) \, dx$ is finite, then $$\lim_{\alpha \to \alpha_0} \int_a^b f(x, \alpha) \, dx = \int_a^b f(x, \alpha_0) \, dx.$$
In other words, is absolute convergence enough to exchange the limit and integration when $f(x, \alpha)$ and $g(x)$ are not continuous at $x = a$ and $x = b$, and $\int_a^b g(x) \, dx < \infty$?
Edit: I think the proposition above follows from a theorem in Introduction to Calculus and Classical Analysis by Hijab (p. 201) or Arzela's dominated convergence theorem (for the Riemann integral). Nevertheless, any input would be greatly appreciated.