I Folland's Real Analysis, I came across to the following theorem,
3.23 Theorem. Let $F: \mathbb{R} \to \mathbb{R} $ be increasing, and let $G(x) = F(x+)$.
The set of points at which $F$ is discontinuous is countable.
$F$ and $G$ are differentiable a.e., and $F'=G'$ a.e.
What does $G(x) = F(x+)$ mean?