Kolk, Enno Convergence-preserving function sequences and uniform convergence. (English) Zbl 0939.40001 J. Math. Anal. Appl. 238, No. 2, 599-603 (1999). 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. Reviewer: R.J.Tomkins (Regina) Cited in 1 ReviewCited in 3 Documents MSC: 40A30 Convergence and divergence of series and sequences of functions 26A03 Foundations: limits and generalizations, elementary topology of the line Keywords:uniform convergence; regular sequence of functions; sequence of real-valued functions; convergence-preserving; convergent sequence PDFBibTeX XMLCite \textit{E. Kolk}, J. Math. Anal. Appl. 238, No. 2, 599--603 (1999; Zbl 0939.40001) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.