Article

Keywords:
order-preserving function; ordered vector space; cone; solid set; continuity
Summary:
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.
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