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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 $G(x) = F(x+)$.

  1. The set of points at which $F$ is discontinuous is countable.

  2. $F$ and $G$ are differentiable a.e., and $F'=G'$ a.e.

What does $G(x) = F(x+)$ mean?

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    It could mean the one-sided limit (approaching $x$ from the right, as in $f(x+0)$). But I'm not sure.2012-12-26
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    You should provide much more context. Which book is this? What was being discussed at this point in the book? What were the preceding and following sentences?2012-12-26
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    This could mean the positive part of $x$, as in $x$ if $x>0$ and $0$ otherwise. Or it could be a typo. Give more context and maybe somebody will have a better idea for you.2012-12-26
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    I added context. thanks!2012-12-26

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It can mean a variety of things:

I am 99% percent sure it is used for: $$f(x+)=\lim_{y\to x^+}f(y)$$ I have seen it used however as: $$f(x+)=\begin{cases} f(x), \ x\in D_f:f(x)>0\\ 0, \ x\in D_f:f(x)<0\end{cases}$$ (we usually use $f_+$ for that) or even in an old extreme book: $$f(x+)=\lim_{h\to 0}\frac{f(x+h)-f(x)}h$$ (we usually use $f^{\prime}_+$ or $f^{\prime}(x+)$ for that). It could of course denote something else like the principle part of $f$ but I doubt it