Consider two compact sets $X$ and $Y$, a function $f:Y\to \mathbb{R}$, and a closed, non-empty correspondence $A:X\twoheadrightarrow Y$. Define the function $G:X\to \mathbb{R}$ by $$ G(x)=\int_{A(x)}f(y)dy. $$ A sufficient conditions for $G$ to be continuous is that $A$ is a continuous correspondence. However, continuity of $A$ is not a necessary condition. For example, let $X=Y=[0,1]$, $f(y)=y$ and $A(x)=\{y\in Y:|y-0.3|\geq x\}$. Then $A(x)$ is upper hemicontinuous but not lower hemicontinuous at $x=0.3$. Yet, $G$ is continuous and given by $$ G(x)=\left\{\begin{array}{ccc}\int^1_{x+0.3}ydy+\int^{0.3-x}_0ydy& \mbox{if}& x\leq 0.3\\\\\int^1_{x+0.3}ydy& \mbox{if}& 0.3<x\leq 0.7\\\\ 0&\mbox{otherwise} \end{array}\right. $$

Does anyone know of a necessary conditions on $A$ that are also sufficient, so that $G$ is a continuous function? (pretty sure upper hemicontinuity alone is not sufficient).