Consider a continuous function $f: X \times Y \rightarrow \mathbb{R}_{} \geq 0$, where $X \subset \mathbb{R}^n$ is compact, and $Y \subseteq \mathbb{R}^m$ is closed.
Define $\hat{f}:X \rightarrow \mathbb{R}_{\geq 0}$ as the parametric integral
$ F(x) \ := \ \int_Y f(x,y) dy $
Assume that $X$ is such that $\sup_{x \in X} F(x) < \infty $.
QUESTION: is $F(\cdot)$ continuous? If not (counterexample please), under which additional conditions we can have continuity?
Examples. $f(x,y)=y^{-x}$ with $Y = [1,\infty)$: $F(x)=1/(x-1)$, $X=[1+\epsilon,M]$, so $\lim_{\delta \rightarrow 0} ( F(x)-F(x+\delta))= \lim_{\delta \rightarrow 0} \frac{\delta}{(x-1)(x+\delta-1)} = 0$. With $f(x,y)=e^{-xy}$ and $Y = [0,\infty)$ we have $F(x) = 1/x$, over $X=[\epsilon,M]$, so $\lim_{\delta \rightarrow 0} ( F(x)-F(x+\delta))=0$ as well. Of course all the cases of the kind $f(x,y)=g(y)h(x)$ with $g(\cdot)$ continuous.