If $\displaystyle f$ is a continuous function on $\displaystyle [a,b]$ such that $\displaystyle \int_{a}^{b}fg = 0$ for all continuous functions $\displaystyle g$. Then $\displaystyle f=0$ on $\displaystyle [a,b]$.

So yeah, I know I have to invoke that $\displaystyle U(fg)=L(fg)=0$ but after that I'm stumped. Thanks.