Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A54

Retrieve articles in all journals with MSC: 26A54


Additional Information

Keywords: La fonction lin&#233;airement-continue, la fonction approximativement lin&#233;airement-continue, l’ensemble du type <!– MATH ${F_\sigma }$ –> <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${F_\sigma }$">, l’ensemble de premi&#232;re categone, le produit cart&#233;sien
Article copyright: © Copyright 1975 American Mathematical Society