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

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.