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The space of density continuous functions. (English) Zbl 0757.26006
The paper deals with density continuous functions, which means functions which are continuous if both the domain and the range are equipped with the density topology. The authors have proved that this class is not closed with respect to the addition and have studied the relations between density continuous functions and some well known classes such as analytic, convex, \(C^ \infty\), and approximately continuous functions. It turned out that convex and analytic functions are density continuous, but there exists a \(C^ \infty\)-function which is not. Obviously each density continuous function is approximately continuous, but there exists a density continuous function which is not continuous.

MSC:
26A21 Classification of real functions; Baire classification of sets and functions
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
26E10 \(C^\infty\)-functions, quasi-analytic functions
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References:
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