Let $f(x,y)$ be a real valued function of two variables, defined for $a
(Can we use Dominated Convergence Theorem to prove this?)
Let $f(x,y)$ be a real valued function of two variables, defined for $a
(Can we use Dominated Convergence Theorem to prove this?)
I remember a similar problem in Measure and Integral by Wheeden and Zygmund. Anyway, the problem becomes simple if you remember the following proposition:
Proposition Let $h : (c, d) \to \Bbb{C}$. Then for each $x_0 \in (c, d)$, $h(x) \to \ell$ as $x \to x_0$ if and only if for every sequences $(x_n) \subset (c, d)$ converging to $x_0$, we have $h(x_n) \to \ell$ as $n \to \infty$.
Now if you consider $h(x) = \int_{a}^{b} f(x, y) \, dy$, then the conclusion is immediate by the Lebesgue's DCT.