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Existence of $x$

Posted: Tue Dec 13, 2016 7:40 pm
by kotsos24919
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

$$\int \limits_A f g = f(x) \int \limits_A g $$

Re: Existence of $x$

Posted: Thu Dec 15, 2016 8:23 am
by S.F.Papadopoulos
What if $ \int _{A}g=0$ and $ \int _{A}fg\neq 0 $ ?