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).



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.