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 ?