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Title: Continuity of order-preserving functions (English)
Author: Lavrič, Boris
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 38
Issue: 4
Year: 1997
Pages: 645-655
Category: math
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. (English)
Keyword: order-preserving function
Keyword: ordered vector space
Keyword: cone
Keyword: solid set
Keyword: continuity
MSC: 26B05
MSC: 26B35
MSC: 47H07
idZBL: Zbl 0942.26022
idMR: MR1601672
Date available: 2009-01-08T18:37:04Z
Last updated: 2012-04-30
Stable URL:
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Reference: [4] Lavrič B.: Continuity of monotone functions.Arch. Math. 29 (1993), 1-4. MR 1242622
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Reference: [6] Stoer J., Witzgall C.: Convexity and Optimization in Finite Dimensions I.Springer-Verlag, Berlin, 1970. MR 0286498


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