TITLE: Does integration preserve continuity?
QUESTION [7 upvotes]: Consider a function $f(x,y):\mathcal{X}\subseteq \mathbb{R}\times \mathcal{Y}\subseteq \mathbb{R}\rightarrow \mathbb{R}$ and suppose that it is continuous over $y$.
Assume also that $\int_{\mathcal{X}} f(x,y) dx$ exists and is finite. 
Can we say that $\int_{\mathcal{X}} f(x,y) dx$ is still continuous over $y$, or I need to impose other restrictions?

REPLY [2 votes]: Here is a simple counterexample:
Take $f(x,y) = |y|\max(0,1-|xy|)$, then $\int f(x,y) dx = \begin{cases} 1 , & y\neq 0 \\ 0, & y=0 \end{cases}$