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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Limit-like predictability for discontinuous functions


Authors: Christopher S. Hardin and Alan D. Taylor
Journal: Proc. Amer. Math. Soc. 137 (2009), 3123-3128
MSC (2000): Primary 03E05; Secondary 54H05, 54C99
Published electronically: March 18, 2009
MathSciNet review: 2506471
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Abstract: Our starting point is the following question: To what extent is a function's value at a point $ x$ of a topological space determined by its values in an arbitrarily small (deleted) neighborhood of $ x$? For continuous functions, the answer is typically ``always'' and the method of prediction of $ f(x)$ is just the limit operator. We generalize this to the case of an arbitrary function mapping a topological space to an arbitrary set. We show that the best one can ever hope to do is to predict correctly except on a scattered set. Moreover, we give a predictor whose error set, in $ T_0$ spaces, is always scattered.


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Additional Information

Christopher S. Hardin
Affiliation: Department of Mathematics and Computer Science, Wabash College, Crawfordsville, Indiana 47933
Email: hardinc@wabash.edu

Alan D. Taylor
Affiliation: Department of Mathematics, Union College, Schenectady, New York 12308
Email: taylora@union.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09877-3
PII: S 0002-9939(09)09877-3
Keywords: Scattered sets
Received by editor(s): June 9, 2008
Received by editor(s) in revised form: January 9, 2009
Published electronically: March 18, 2009
Communicated by: Julia Knight
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.