Existence of $x$
Posted: Tue Dec 13, 2016 7:40 pm
Let $A \subseteq \mathbb{R}^n$ be a compact and connected subset of $\mathbb{R}^n$. Suppose that $f:A \rightarrow \mathbb{R}$ is a continuous function and $g:A \rightarrow \mathbb{R}$ an integrable one. Prove that there exists an $x$ such that
$$\int \limits_A f g = f(x) \int \limits_A g $$
$$\int \limits_A f g = f(x) \int \limits_A g $$