Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Existence of $x$

Real Analysis
kotsos24919
Articles: 0
Posts: 10
Joined: Mon May 30, 2016 9:19 pm

### Existence of $x$

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