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Existence of $x$

Real Analysis
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kotsos24919
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Existence of $x$

#1

Post by kotsos24919 » 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

$$\int \limits_A f g = f(x) \int \limits_A g $$
S.F.Papadopoulos
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Re: Existence of $x$

#2

Post by S.F.Papadopoulos » Thu Dec 15, 2016 8:23 am

What if $ \int _{A}g=0$ and $ \int _{A}fg\neq 0 $ ?
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