I wonder it is possible to extend the mean value theorem not on compactness.
In more detail, Let $f : A \rightarrow \mathbb{R}$ be continuous on $A \subset \mathbb{R}^n$. The mean value theorem for integral states that if $A$ is connected and compact, there exists $x_{0} \in A$ such that $$ \frac{1}{\nu(A)}\int_{A}f(x)dx = f(x_{0}) \;\;\; (\nu(A)\text{ is a volume of A}) $$
Now, the point I am curious is that if I can generalize the mean value theorem for integrals to the case $A$ is not compact, just bounded and connected.
I think it is possible, if I can take the supremum and infimum of $f(x)$, which is treaky for me.
Could anybody help to extend the idea? Thanks in advance.