We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.
References
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Godement, R. (2004) Analysis I. Convergence, Elementary Functions. Springer, Berlin.
Ross, K.A. (2013) Elementary Analysis. The Theory of Calculus. Springer, New York.
http://dx.doi.org/10.1007/978-1-4614-6271-2
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Godement, R. and Spain, P. (2005) Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Fnctions. Springer, Berlin.
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Ezzinbi, K., Degla, G. and Ndambomve, P. (in Press) Controllability for Some Partial Functional Integrodifferential Equations with Nonlocal Conditions in Banach Spaces. Discussiones Mathematicae Differential Inclusions Control and Optimization.
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Freslon, J., Poineau, J., Fredon, D. and Morin, C. (2010) Mathématiques. Exercices Incontournables MP. Dunod, Paris.
