Natural function

Definition

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 $E_a^b$.

Some characterizations of natural functions

Theorem. 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 $V(f,E)=\sup(E)-\inf(E)$.

Theorem. Let $X \neq\{0\}$ be a normed space over $\mathbb{R}$ and let $f:[a,b] \to X$ be a function with positive and bounded variation. Then the following conditions are equivalent:

(a) $f/\alpha$ is a natural function, where $\alpha = V(f,[a,b])/(b-a)$;

(b) $f$ is continuous at points $a$ and $b$, and there exists a set $D$ dense in $[a,b]$ such that

$(\forall t,s,p,q \in D) (t<s, p<q, s-t \leq q-p) => V(f,[t,s]) \leq V(f,[p,q])$;

(c) $f$ is continuous at points $a$ and $b$, and for every set $D$ dense in $[a,b]$ such that

$(\forall t,s,p,q \in D) (t<s, p<q, s-t \leq q-p) => V(f,[t,s]) \leq V(f,[p,q])$;

(d) $(\forall t,s,p,q \in [a,b]) (t<s, p<q, s-t = q-p) => V(f,[t,s]) = V(f,[p,q])$.

Theorem. Let $f:[a,b] \to \mathbb{R}$. Then $f$ is a natural function if and only if $f$ is absolutely continuous and $|f'(x)|=1$ a.e. on $[a,b]$.

Theorem. 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.

[2] M. Małolepszy, On natural functions and Lipschitz functions, Real Analysis Exchange, Vol. 28(1), 2002/2003, 255-264.