Continuity of order-preserving functions

Boris Lavrič

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 4, page 645-655
  • ISSN: 0010-2628

Abstract

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Let the spaces 𝐑 m and 𝐑 n be ordered by cones P and Q respectively, let A be a nonempty subset of 𝐑 m , and let f : A 𝐑 n be an order-preserving function. Suppose that P is generating in 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 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.

How to cite

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Lavrič, Boris. "Continuity of order-preserving functions." Commentationes Mathematicae Universitatis Carolinae 38.4 (1997): 645-655. <http://eudml.org/doc/248078>.

@article{Lavrič1997,
abstract = {Let the spaces $\mathbf \{R\}^m$ and $\mathbf \{R\}^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of $\mathbf \{R\}^m$, and let $f:A\longrightarrow \mathbf \{R\}^n$ be an order-preserving function. Suppose that $P$ is generating in $\mathbf \{R\}^m$, 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 $\mathbf \{R\}^m$. 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.},
author = {Lavrič, Boris},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {order-preserving function; ordered vector space; cone; solid set; continuity; ordered vector spaces; cone; solid set; order-preserving function; boundedness; continuity},
language = {eng},
number = {4},
pages = {645-655},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Continuity of order-preserving functions},
url = {http://eudml.org/doc/248078},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Lavrič, Boris
TI - Continuity of order-preserving functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 4
SP - 645
EP - 655
AB - Let the spaces $\mathbf {R}^m$ and $\mathbf {R}^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of $\mathbf {R}^m$, and let $f:A\longrightarrow \mathbf {R}^n$ be an order-preserving function. Suppose that $P$ is generating in $\mathbf {R}^m$, 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 $\mathbf {R}^m$. 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.
LA - eng
KW - order-preserving function; ordered vector space; cone; solid set; continuity; ordered vector spaces; cone; solid set; order-preserving function; boundedness; continuity
UR - http://eudml.org/doc/248078
ER -

References

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  1. Debreu G., Continuity properties of Paretian utility, Internat. Econom. Rev. 5 (1964), 285-293. (1964) 
  2. Fishburn P.C., Utility Theory for Decision Making, J. Wiley and Sons, New York, London, Sidney, Toronto, 1970. MR0264810
  3. Jameson G., Ordered linear spaces, Lecture Notes in Math., Vol. 141, Springer-Verlag, Berlin, Heidelberg, New York, 1970. MR0438077
  4. Lavrič B., Continuity of monotone functions, Arch. Math. 29 (1993), 1-4. (1993) MR1242622
  5. Rockafellar R.T., Convex Analysis, Princeton Univ. Press, Princeton, N.J., 1972. MR1451876
  6. Stoer J., Witzgall C., Convexity and Optimization in Finite Dimensions I, Springer-Verlag, Berlin, 1970. MR0286498

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