Limit of integral - part 2 Inspired by the recent post "Limit of integral", I propose the following problem (hoping it will not turn out to be too easy). Suppose that $g:[0,1] \times [0,1] \to {\bf R}$ is continuous in both variables separately. Is it true that, for all $x_0 \in [0,1]$,
$$
 \lim \limits_{x \to x_0 } \int_0^1 {g(x,y)\,{\rm d}y}  = \int_0^1 {g(x_0 ,y)\,{\rm d}y} .
$$ 

This is true if there is a (measurable) function $f(y)$ such that $$|g(x,y)|\le f(y), \mbox{ for }0\le x\le 1 $$  and $$\int_0^1 f(y)\,dy<\infty.$$
This follows from the Lebesgue dominated convergence theorem.
Note: Every continuous function is measurable.
