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Convergence-preserving function sequences and uniform convergence. (English) Zbl 0939.40001

Consider a sequence of real-valued functions \(\{f_k, k\geq 1\}\) defined on a closed subset \(S\) of the real line. This sequence is called convergence-preserving on \(S\) if \(\{f_k(t_k), k\geq 1\}\) is convergent for every convergent sequence \(\{t_k\} \subset S\). The main result shows that \(\{f_k\}\) is convergence-preserving on a closed interval \([a,b]\) if and only if the sequence \(\{f_k\}\) converges uniformly on \([a,b]\) to a continuous function.

MSC:

40A30 Convergence and divergence of series and sequences of functions
26A03 Foundations: limits and generalizations, elementary topology of the line
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References:

[1] Wilansky, A., Summability through Functional Analysis (1984), North-Holland: North-Holland Amsterdam/New York, Oxford · Zbl 0531.40008
[2] Wildenberg, G., Convergence-preserving functions, Amer. Math. Monthly, 95, 542-544 (1988) · Zbl 0664.40001
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