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 ?⊂ℝ? such that at each ?0∈? the “thick” limit ??0(?)=lim?→0+?(?0+??), exists for all ?∈? , the unit sphere of ℝ? . We study the set of singular points of f, namely, the set of points ? where the thick limit is not constant. In one variable it is well known that ? is countable. We give examples where ? is not countable in ℝ? , but we prove that if all the thick values are continuous functions of w, then ? must be countable. We also consider regulated distributions, elements of the space ′(?) 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 ? as well.