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When the identity theorem "seems" to fail

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Conejero, Jose A. and Jiménez Rodríguez, P. and Muñoz-Fernández, Gustavo A. and Seoane-Sepúlveda, Juan B. (2014) When the identity theorem "seems" to fail. American Mathematical Monthly, 121 (1). pp. 60-68. ISSN 0002-9890

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Official URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.121.01.060


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Abstract

The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting, if we relax the analytic hypothesis on the function to infinitely many times differentiable. In fact, we construct an algebra of functions A enjoying the following properties: (i) A is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of A is uncountable); (ii) every nonzero element of A is nowhere analytic; (iii) A subset of C-infinity (R); (iv) every element of A has infinitely many zeros in R; and (v) for every f is an element of A\ {0} and n is an element of N, f((n)) (the nth derivative of f) enjoys the same properties as the elements in A\ {0}.
This construction complements those made by Cater and by Kim and Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively.


Item Type:Article
Uncontrolled Keywords:Nowhere analytic-functions; lineability; algebrability; sets
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:24707
Deposited On:17 Mar 2014 12:54
Last Modified:28 Nov 2016 09:36

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