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.