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.
Best Answer
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!)