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I'm trying to solve a classical problem on right continuous functions: Every right continuous increasing function $ F$ is the sum of a continuous function $C$ and a jump function $J$ (pice-wise constant).

I´ve already prooved that both sided limits exists in every point and that the set of discontinuities is countable. My idea is to define $ H$ to be constant in each open interval in which $ F$ is continuous and equal to the difference between the right and left handed limit. To do that I would like to know that $D$, the set of discontinuities, is discrete or at least closed (which seems true for me) but I couldn't. I've been trying to proove that in a limit point of $ D$, $ F$ cannot be right continuous or something like that but with not success. I would appriciate any help and thank you in advance :)

Pd: I'm not asking for the solution of the first problem, I just want to know if my idea works and if it does have some help, so please do not show me solutions of the first problem.

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    $\begingroup$ No, $D$ is in general not closed. Consider for instance $$F(x) := \sum_{n \geq 1} \frac{1}{2^n} 1_{[2^{-n-1},2^{-n})}(x),$$ then $F$ is continuous at $x=0$ and there exists a sequence of discontinuity points converging to $x=0$. $\endgroup$
    – saz
    Feb 4, 2019 at 20:43
  • $\begingroup$ I see, thank you. $\endgroup$
    – Natalio
    Feb 4, 2019 at 20:50

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There is misconception: A jump function is not necessarily piecewise constant, see e.g. saz's example. This function is not constant in any interval $[0,\varepsilon]$, but a "jump function" according to the usal Lebesgue decomposition.

Hint for the problem: You know that the set of all discontinuities is (at most) countable. Thus, let $(y_n)_{n \in \mathbb{N}}$ be an enumeration of the set of all discontinuities. Define the "jump height" by $$H(x) := \lim_{h \downarrow x} F(h) - \lim_{h \uparrow x} F(h) = F(x) - \lim_{h \downarrow x} F(x).$$ Note that $H(y_n) >0$. Now construct a function which is piecewise constant and has exactly jumps in $y_n$ with height $H(y_n)$. (saz's example may help you!)

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