The set of singularities of regulated functions in several variables

• Autores: Ricardo Estrada
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 63, Fasc. 3, 2012, págs. 351-359
• Idioma: inglés
• DOI: 10.1007/s13348-011-0042-z
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• Resumen

  • 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.