Abstract: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.
Keywords: order-preserving function, ordered vector space, cone, solid set, continuity
AMS Subject Classification: 26B05, 47H07