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Let $f(x,y)$ be a real valued function of two variables, defined for $a<y<b, c<x<d$. 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?)

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How about: $\lim_{x \to x_0} \phi(x) = L$ if and only if for every sequence $x_n$ with $x_n \to x_0$ we have $\lim_{n \to \infty} \phi(x_n) = L$. –  GEdgar Oct 23 '12 at 16:59
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Let me suggest you learn how to accept answers. –  Did Oct 23 '12 at 18:47
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