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We show that over closed bounded intervals in certain Archimedean
ordered fields as well as in all non-Archimedean ones of countable
cofinality, there are {\it uniformly continuous} 1-1 functions not
mapping interior to interior. For the latter kind of fields, there
are also uniformly continuous 1-1 functions mapping all interior
points to interior points of the image which are, nevertheless,
not open. In particular the ordered Laurent and Puiseux series
fields with coefficients in any ordered field accommodate both
kinds of such strange functions.
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