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I have a question on a variation of the fundamental lemma .

If $\int_\Omega f(x) g(x)=0$ and $f, g $ are $C^0\Omega$ functions and $\int_\Omega g(x)=0 $

then is it possible that there exist some constant $c$ such that $f(x)=c$ for all $x\in \Omega$

I tried to use mollification on one of the function and throw derivative on the mollified function but that doesn't give me anything . I am wondering if the question makes sense ?

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    I don't understand: yes it's possible that $f$ is constant (nothing prevents it).2012-10-31
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    @DavideGiraudo : I am trying to prove it but i am not able to do it . Can you give me some guidelines to start off with proof . .2012-10-31
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    One should specify the hypothesis of the lemma. Suppose that $\Omega$ is a bounded interval $(a,b)$ and let $f \in C^0({\overline{\Omega}})$ verify $\int_\Omega f(x) g(x)=0$ for all $g \in C^0({\overline{\Omega}})$ such that $\int_\Omega g(x)=0$. Then there exists a constant $c \in \mathbb{R}$, such that $f(x)=c$ for all $x \in \overline{\Omega}$.2017-01-31
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    It seems that the correct interpretation of the problem is the one given by H.H.Rugh below. If this is indeed the case, you should edit your question and reformulate, otherwise somebody might waste time answering the wrong question.2017-02-04

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