Let $f : \Bbb R \longrightarrow \Bbb R$ be a continuous function and $g : \Bbb R \longrightarrow \Bbb R$ be a function such that $g = f$ almost everywhere. Can we say that $g$ is continuous almost everywhere?
I think it is true because by the given condition there exists a subset $E$ of $\Bbb R$ with measure $0$ such that $g(x) = f(x),$ for all $x \in \Bbb R \setminus E.$
Any help in this regard will be highly appreciated. Thank you very much for your valuable time.
No. If $f(x)=0$ for all $x$, $g(x)=0$ for $x$ irrational and $1$ for $x$ rational then $f=g$ almost everywhere but $g$ is not continuous at any point.