Continuity of order-preserving functions

Lavrič, Boris. "Continuity of order-preserving functions." Commentationes Mathematicae Universitatis Carolinae 38.4 (1997): 645-655. <http://eudml.org/doc/248078>.

@article{Lavrič1997,
abstract = {Let the spaces $\mathbf \{R\}^m$ and $\mathbf \{R\}^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of $\mathbf \{R\}^m$, and let $f:A\longrightarrow \mathbf \{R\}^n$ be an order-preserving function. Suppose that $P$ is generating in $\mathbf \{R\}^m$, and that $Q$ contains no affine line. Then $f$ is locally bounded on the interior of $A$, and continuous almost everywhere with respect to the Lebesgue measure on $\mathbf \{R\}^m$. If in addition $P$ is a closed halfspace and if $A$ is connected, then $f$ is continuous if and only if the range $f(A)$ is connected.},
author = {Lavrič, Boris},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {order-preserving function; ordered vector space; cone; solid set; continuity; ordered vector spaces; cone; solid set; order-preserving function; boundedness; continuity},
language = {eng},
number = {4},
pages = {645-655},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Continuity of order-preserving functions},
url = {http://eudml.org/doc/248078},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Lavrič, Boris
TI - Continuity of order-preserving functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 4
SP - 645
EP - 655
AB - Let the spaces $\mathbf {R}^m$ and $\mathbf {R}^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of $\mathbf {R}^m$, and let $f:A\longrightarrow \mathbf {R}^n$ be an order-preserving function. Suppose that $P$ is generating in $\mathbf {R}^m$, and that $Q$ contains no affine line. Then $f$ is locally bounded on the interior of $A$, and continuous almost everywhere with respect to the Lebesgue measure on $\mathbf {R}^m$. If in addition $P$ is a closed halfspace and if $A$ is connected, then $f$ is continuous if and only if the range $f(A)$ is connected.
LA - eng
KW - order-preserving function; ordered vector space; cone; solid set; continuity; ordered vector spaces; cone; solid set; order-preserving function; boundedness; continuity
UR - http://eudml.org/doc/248078
ER -