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Journal of the London Mathematical Society 2001 64(1):81-92; doi:10.1017/S0024610701002265
© 2001 by London Mathematical Society
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© The London Mathematical Society

An Access Theorem for Continuous Functions

Alexander Borichev and Igor Kleschevich

Department of Mathematics, University of Bordeaux I 351 cours de la Liberation, 33405 Talence, France, borichev{at}math.u-bordeaux.fr
19 Mount Hood Road 4, Brighton, MA 02215, USA

Received 6 January 1999. Revision received 18 October 2000.

Let f be a continuous function on an open subset {Omega} of R2 such that for every x isin {Omega} there exists a continuous map {gamma} : [–1, 1] -> {Omega} with {gamma}(0) = x and f {circ} {gamma} increasing on [–1, 1]. Then for every {gamma} isin {Omega} there exists a continuous map {gamma} : [0, 1) -> {Omega} such that {gamma}(0) = y, f {circ} {gamma} is increasing on [0; 1), and for every compact subset K of {Omega}, max{t : {gamma}(t) isin K} < 1. This result gives an answer to a question posed by M. Ortel. Furthermore, an example shows that this result is not valid in higher dimensions.


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