Uniform limits of preponderantly continuous functions. (English) Zbl 1278.26003

Let \(\lambda\) be the Lebesgue measure on the real line. A real number \(x_0\) is a point of preponderant density in Denjoy’s sense of a measurable set \(E \subseteq R\) if \(\liminf_{\lambda(J) \to 0, x_0 \in J}\frac{\lambda(J\cap E)}{\lambda(J)} > \frac 12\). A function \(f:I \to R\) is preponderantly continuous at \(x_0 \in I\) in Denjoy’s sense if there is a measurable set \(E\) containing \(x_0\) such that the restriction of \(f\) to \(E\) is continuous at \(x_0\) and \(x_0\) is a point of preponderant density of \(E\). A point \(x_0\) is a point of preponderant density in O’Malley’s sense of \(E\) if there is a closed neighborhood of \(x_0\) such that each of its closed subintervals \(J\) containing \(x_0\) fulfills the inequality \(\frac{\lambda(E\cap J)}{\lambda(J)} > \frac 12\). Preponderant continuity in O’Malley’s sense is defined analogously. It is shown that while preponderant continuity in O’Malley’s sense is preserved under uniform limit, it is not the case for the Denjoy preponderantly continuous functions. The metric and topological properties of these families of functions are studied.


26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C30 Real-valued functions in general topology
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