Jump functions

The simplest of Monotonic functions are called jump functions, they are defined in the following way: Let $I\subseteq \mathbb{R}$ be an interval, suppose we have a finite or countable collection of points $x_1,x_n,\dots$ in this interval, for each point we assign a value $h_n$ such that ${\sum }_n{h}_n<\infty$ and $h_n>0$ for all $n$.

$$f(x):=\sum _{n:x_n<x}{h}_n$$

where we define $f(x)=0$ is the sum has zero terms. This function is:

1. Monotonic non-decreasing
2. Left-continuous¹, with jump discontinuities at every $x_n$ with corresponding jump value $h_n$, continuous at every other point.

Example

We can construct a Jump function that is has jumps in a dense set. Let $\{x_n\}=\mathbb{Q}$ (with a choosen denumeration) and $h_n$ be any positive sequence in $l^1$, for example $h_n=\frac{1}{2^n}$.

Footnotes §

1. If we insted defined $f$ as

   $$f(x)=\sum _{n:x_n\le x}{h}_n$$

   then it would be right-continuous. ↩