Natural function

A function $g: E \to X$ is natural if $V(g,E_a^b)=b-a$ for all $a,b \in E$, $a \leq b $, where $E\subset \mathbb{R}$ is a non-empty bounded set, $E_a^b=\{s \in E: a \leq s \leq b \}$ for $a,b \in E$ ($a \leq b$), $X$ is a metric space with a metric $d$, $V(g,E_a^b)$ variation of $g$ on $E_a^b$.

Let ${E_t}^-=\{s \in E: s \leq t\}$ and ${E_t}^+=\{s \in E: t \leq s\}$. The following conditions are equivalent:

(a) $f$ is a natural function;

(b) $V(f,E_x^-)=x+c$, $x \in E$, where $c=-inf(E)$;

(c) $f$ is a Lipschitz function such that $Lip(f) \leq 1$ and $V(f,E)=sup(E)-inf(E)$.

A natural function is a Lipschitz function and the smallest Lipschitz constant equals $1$.

A function $f: E \to X$ has bounded variation if and only if there exists a non-decreasing bounded function $\phi : E \to \mathbb{R}$ and a natural function $g: \phi (E) \to X$ such that $f=g\circ\phi$ on $E$.

References

[1] V.V. Chistyakov, On the theory of set-valued maps of bounded variation of one real variable, Sbornik: Mathematics 189:5 (1998), 797-819.