A variation of fundamental lemma of variation of calculus .

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). – Davide Giraudo Oct 31 '12 at 10:44
@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 . . – Theorem Oct 31 '12 at 20:56