Let $f$ be a real-valued function of two variables $(x,y)$ that is defined on the square $Q=\{(x,y)| 0 \leq x \leq 1, 0 \leq y \leq 1\}$ and is a measurable function of x for each fixed value of y. Suppose for each fixed value of x, $\lim_{y \to 0} f(x,y) = f(x)$ and that for all y, we have $|f(x,y)| \leq g(x)$, where g is integrable over [0,1]. Show that
$\lim_{y \to 0} $$\int_{0}^{1} f(x,y) dx$$ = $$\int_{0}^{1} f(x) dx$$ $
Also show that if the function f(x,y) is continuous in y for each x, then
$h(y)= $$\int_{0}^{1} f(x,y) dx$$ $ is a continuous function of y
Do you know Dominated Convergence Theorem? Both parts of your question are immediate applications of this theorem.