Let $f$ be an integrable function in $(0,1)$. Suppose that $$\int_{0}^{1} fg dx \geq 0$$ for any nonnegative continuous function $g:(0,1) \to \mathbb{R}$. Prove that $f \geq 0$ almost everywhere in $(0,1)$.
I've tried to approach this problem via contrapositive, and build a function $g$ to make this fail, where $g$ is large whenever $f$ is negative, but I don't know how to accomplish that while making sure $g$ is continuous.