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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Une caractérisation des ensembles des points de discontinuité des fonctions linéairement-continues


Author: Zbigniew Grande
Journal: Proc. Amer. Math. Soc. 52 (1975), 257-262
MSC: Primary 26A54
DOI: https://doi.org/10.1090/S0002-9939-1975-0374349-0
MathSciNet review: 0374349
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Abstract: A function $ f:{R^2} \to R$ (where $ R$ is the set of real numbers) is called linearly-continuous if for each $ x$ and $ y$ the functions $ {f_x}$ and $ {f^y}$ given by $ {f_x}(t) = f(x,t)$ and $ {f^y}(t) = f(t,y)$ for $ - \infty < t < \infty $ are continuous. It is proven that: A set $ A \subset {R^2}$ is the set of points of discontinuity for a linearly-continuous function iff $ A$ is $ {F_\sigma }$ contained in a cartesian product of two linear sets of first category. It is proven also that an analogous characterisation is not possible for an approximatively linearly-continuous function.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0374349-0
Keywords: La fonction linéairement-continue, la fonction approximativement linéairement-continue, l'ensemble du type $ {F_\sigma }$, l'ensemble de première categone, le produit cartésien
Article copyright: © Copyright 1975 American Mathematical Society