The jump function of a nondecreasing function on a compact interval

Let $a\le b$, and let $F:[a,b]\to \mathbb{R}$ be nondecreasing (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences). When $a<b$, write

${\beta }_a:={\lim }_{x\to a^{+}}F(x)-F(a),$

which exists by One-sided limits of a monotone function always exist: for $f$ nondecreasing on an interval $I$ and $c\in I$, ${\lim }_{x\to c^{-}}f(x)=\sup \{f(x):x\in I,x<c\}$ whenever $I$ has points below $c$, ${\lim }_{x\to c^{+}}f(x)=\inf \{f(x):x\in I,x>c\}$ whenever it has points above $c$, and these satisfy ${\lim }_{x\to c^{-}}f(x)\le f(c)\le {\lim }_{x\to c^{+}}f(x)$ and is nonnegative. When $a=b$, put ${\beta }_a:=0$. For $x\in [a,b]$ define the jump function $J_F$ by

$J_F(a):=0,$

and, for $x>a$,

$J_F(x):={\beta }_a+\sup \left\{{\sum }_{t\in S}\right(F(t)-F(t^{-}))+{\sum }_{u\in T}(F(u^{+})-F(u)):\begin{array}{l}S\subseteq (a,x]\text{ finite},\\ T\subseteq (a,x)\text{ finite}\end{array}\}\mathrm{.}$

The summands are nonnegative, and the supremum is taken in the complete ordered field of the reals (Complete ordered field (least-upper-bound property)).

• When $a<b$, the endpoint defect at $a$ is included separately because a nondecreasing function on $[a,b]$ can fail to be continuous at the left endpoint without having a left-hand jump there. The convention ${\beta }_a=0$ makes the degenerate interval $[a,a]$ harmless.
• The first sum collects left jumps at points at or before $x$, while the second collects right jumps at points strictly before $x$. Later A nondecreasing function splits uniquely into a jump part and a continuous part ↗ proves that these two contributions exactly remove the discontinuities of $F$, so $F-J_F$ is continuous.
• If $F$ is right-continuous, then every interior right jump is zero and the definition reduces to the usual cumulative left-jump function.