If $f=g$ a.e. and $f$ and $g$ are continuous, then $f=g$ every where. Let $f,g:\mathbb R\longrightarrow \mathbb R$ two continuous functions. Is it true that if $f=g$ a.e. then $f=g$ every where ? I would say yes. My argument would be the following one : Let $$\Omega =\\{x\mid f(x)\neq g(x)\\}.$$ So let $x\in \Omega $. Is it true that there is a sequence in $\mathbb R\backslash \Omega $ that converge to $x$ ? If yes, the claim is proved. But if yes, How can I prove this claim ? If no, how can I do ? Does this result can be adapted to an abstracted measurable space $(X,\mu)$ ?

Since $f$ and $g$ are continuous, $\Omega$ is open, if it is not empty, its measure is not zero.