# The set of discontinuities of an increasing right continuous function is closed.

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.

• 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 $x=0$.
– saz
Feb 4, 2019 at 20:43
• I see, thank you. Feb 4, 2019 at 20:50

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!)