The set of singularities of regulated functions in several variables

Abstract

We consider a class of regulated functions of several variables, namely, the class of functions f defined in an open set ${U\subset\mathbb{R}^{n}}$ such that at each ${{\bf x}_{0} \in U}$ the “thick” limit

$$ f_{{\bf x}_{0}}\left( {\bf w}\right) =\lim_{\varepsilon\rightarrow 0^{+}}f\left( {\bf x}_{0}+\varepsilon{\bf w}\right), $$

exists for all ${{\bf w}\in\mathbb{S}}$, the unit sphere of ${\mathbb{R}^{n}}$. We study the set of singular points of f, namely, the set of points ${\mathfrak{S}}$ where the thick limit is not constant. In one variable it is well known that ${\mathfrak{S}}$ is countable. We give examples where ${\mathfrak{S}}$ is not countable in ${\mathbb{R}^{n}}$, but we prove that if all the thick values are continuous functions of w, then ${\mathfrak{S}}$ must be countable. We also consider regulated distributions, elements of the space ${\mathcal{D}^{\prime} \left(U\right)}$ for which the thick value exists, as a distributional limit, and show that in this case the continuity of the thick values gives the countability of ${\mathfrak{S}}$ as well.