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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A pointwise convergence theorem for sequences of continuous functions.

Author: K. Schrader
Journal: Trans. Amer. Math. Soc. 159 (1971), 155-163
MSC: Primary 40.20
MathSciNet review: 0280902
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Abstract: Let $ \{ {f_k}\} $ be a sequence of continuous real valued functions defined on an interval $ I$ and $ N$ a fixed nonnegative integer such that if $ {f_k}(x) = {f_i}(x)$ for more than $ N$ distinct values of $ x \in I$ then $ {f_{k}}(x) \equiv {f_i}(x)$ for $ x \in I$. It follows that there is a subsequence $ \{ {g_j}\} $ of $ \{ {f_k}\} $ such that for each $ x$ the subsequence $ \{ {g_j}(x)\} $ is eventually monotone. Thus $ {\lim _{j \to + \infty }}{g_j}(x) = f(x)$ exists for all $ x$, where $ f$ is an extended real valued function. If $ \vert{f_k}(x)\vert$ is bounded for each $ x \in I$ then $ {\lim _{j \to + \infty }}{g_j}(x) = f(x)$ exists as a finite limit for all $ x \in I$. For $ N = 0$ this reduces to picking a monotone subsequence from a sequence of continuous functions whose graphs are pairwise disjoint.

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Keywords: Sequences of functions, pointwise convergence, boundary value problems, uniqueness of solutions
Article copyright: © Copyright 1971 American Mathematical Society

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