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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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    I added context. thanks!2012-12-26

1 Answers 1

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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