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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Une caractérisation des ensembles des points de discontinuité des fonctions linéairement-continues
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by Zbigniew Grande PDF
Proc. Amer. Math. Soc. 52 (1975), 257-262 Request permission

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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • 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