Geodesic
Continuity of order-preserving functions
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 4, pp. 645-655 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let the spaces $\bold R^m$ and $\bold R^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of $\bold R^m$, and let $f:A\longrightarrow \bold R^n$ be an order-preserving function. Suppose that $P$ is generating in $\bold R^m$, 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 $\bold R^m$. 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.
Let the spaces $\bold R^m$ and $\bold R^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of $\bold R^m$, and let $f:A\longrightarrow \bold R^n$ be an order-preserving function. Suppose that $P$ is generating in $\bold R^m$, 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 $\bold R^m$. 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.
Classification : 26B05, 26B35, 47H07
Keywords: order-preserving function; ordered vector space; cone; solid set; continuity 
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     author = {Lavri\v{c}, Boris},
     title = {Continuity of order-preserving functions},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {645--655},
     year = {1997},
     volume = {38},
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     zbl = {0942.26022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1997_38_4_a3/}
}
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Lavrič, Boris. Continuity of order-preserving functions. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) no. 4, pp. 645-655. http://geodesic.mathdoc.fr/item/CMUC_1997_38_4_a3/