Difference between revisions of "Natural function"
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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 $E_a^b$. | 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 $E_a^b$. | ||
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| + | Let $E_t^-=\{s \in E: s \leq t\}$ | ||
A natural function is a Lipschitz function and the smallest Lipschitz constant equals $1$. | A natural function is a Lipschitz function and the smallest Lipschitz constant equals $1$. | ||
Revision as of 16:09, 21 April 2016
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.
Natural function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_function&oldid=27387