We are doing a problem that requires us to, for fixed NONZERO element $h \in L^2[0,1]$ (as in, the function $h$ is not zero on a set of positive measure in $[0,1]$) and fixed $f \in L^2[0,1]$ with the property that $\int fh =0$, and fixed $\epsilon >0$, find a function $g \in L^2[0,1]$ such that $||f-g||_2<\epsilon$ but $\int hg \neq 0$.
This has proven to be difficult, though intuitively it makes sense. We tried several hatting ideas, and using the fact that $h^2$ must have a nonzero integral, to no resolution. Any ideas?