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, $V(g,E_a^b)$ variation of $g$ on $E_a^b$.

Let $E_t^{-}=\{s \in E: s \leq t\}$

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.