order-preserving function; ordered vector space; cone; solid set; continuity
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.
 Debreu G.: Continuity properties of Paretian utility. Internat. Econom. Rev. 5 (1964), 285-293.
 Fishburn P.C.: Utility Theory for Decision Making
. J. Wiley and Sons, New York, London, Sidney, Toronto, 1970. MR 0264810
 Jameson G.: Ordered linear spaces
. Lecture Notes in Math., Vol. 141, Springer-Verlag, Berlin, Heidelberg, New York, 1970. MR 0438077
 Lavrič B.: Continuity of monotone functions
. Arch. Math. 29 (1993), 1-4. MR 1242622
 Rockafellar R.T.: Convex Analysis
. Princeton Univ. Press, Princeton, N.J., 1972. MR 1451876
 Stoer J., Witzgall C.: Convexity and Optimization in Finite Dimensions I
. Springer-Verlag, Berlin, 1970. MR 0286498