Let $f(x,y)$ be a real valued function of two variables, defined for $a. Assume that for each $x, f(x,.)$ is a Borel measurable function of $y$, and that there is a Borel measurable function $g: (a,b)\to$ $\Bbb R$ such that $|f(x,y)|<= g(y)$ for all $x,y $ and $\int_a^bg(y) dy <\infty.$ If $x_0\in (c,d)$ and $\lim_{x \to x_0}f(x,y)$ exist for all $y\in (a,b)$, show that $\lim_{x \to x_0}\int_a^bf(x,y) dy = \int_a^b[\lim_{x \to x_0} f(x,y)]dy $
(Can we use Dominated Convergence Theorem to prove this?)
real-analysis
integration