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.
@article{118963,
author = {Boris Lavri\v c},
title = {Continuity of order-preserving functions},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {38},
year = {1997},
pages = {645-655},
zbl = {0942.26022},
mrnumber = {1601672},
language = {en},
url = {http://dml.mathdoc.fr/item/118963}
}
Lavrič, Boris. Continuity of order-preserving functions. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 645-655. http://gdmltest.u-ga.fr/item/118963/
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