The set of singularities of regulated functions in several variables

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We consider a class of regulated functions of several variables, namely, the class of functions f defined in an open set U ⊂ ℝ n such that at each x 0 ∈ U the "thick" limit, exists for all w ∈ S, the unit sphere of ℝ n. We study the set of singular points of f, namely, the set of points S where the thick limit is not constant. In one variable it is well known that S is countable. We give examples where S is not countable in ℝ n, but we prove that if all the thick values are continuous functions of w, then S must be countable. We also consider regulated distributions, elements of the space D′ (U) 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 S as well. © 2011 Universitat de Barcelona.

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Collectanea Mathematica

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