This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in C^{\infty}_{o}(\Omega)$. Then by du Boise-Reymond lemma we have that $f = 0$ a.e. Can anyone show that we can further show that $f = 0$ everywhere? This is a link to the definition of du Boise-Reymond lemma.
Thanks.
If a continuous function $f$ satisfies $f=0$ on a dense subset of $\Omega$, then $f=0$ everywhere in $\Omega$. This follows from any definition of continuity you may be using: with limits or open sets. A set whose complement has zero measure is dense.