Counting Path and Its Jump Times

Let $\mathbb{R}$ be the set of real numbers. A function $c:[0,\infty)\to\mathbb{R}$ is a counting path if:

1. (Integer values.) $c(0)=0$ and, for every $t\ge 0$, $c(t)$ is either $0$ or a natural number.

2. (Monotonicity.) $c(s)\le c(t)$ whenever $0\le s\le t$.

3. (Right-continuity.) For every $t\ge 0$, $c(t)$ is the greatest lower bound of the set $\{c(s):s>t\}$.

4. (Unit jumps.) For $t>0$ write $c(t-)$ for the least upper bound of the set $\{c(s):0\le s<t\}$, and set $c(0-)=0$. Then $c(t)-c(t-)\le 1$ for every $t\ge 0$.

A jump time of a counting path $c$ is a real number $t>0$ with $c(t)>c(t-)$. For a natural number $k\ge 1$, the $k$-th jump time of $c$ is

the greatest lower bound of the displayed set when it is nonempty, with the convention $\tau_k(c)=+\infty$ when the set is empty.