The space of density continuous functions.

*(English)*Zbl 0757.26006The 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.

Reviewer: J.Niewiarowski (Łódź)

##### 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 |

##### Keywords:

convex functions; \(C^ \infty\)-function; density continuous functions; density topology; approximately continuous functions; analytic functions
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\textit{K. Ciesielski} and \textit{L. Larson}, Acta Math. Hung. 58, No. 3--4, 289--296 (1991; Zbl 0757.26006)

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##### References:

[1] | A. M. Bruckner, Density-preserving homeomorphisms and a theorem of Maximoff,Quart. J. Math. Oxford,21 (1970), 337–347. · Zbl 0196.07701 · doi:10.1093/qmath/21.3.337 |

[2] | I. P. Natanson,Theory of Functions of a Real Variable, Vol. 2, Frederick Ungar Publishing Co. (New York, 1964). · Zbl 0156.29202 |

[3] | Jerzy Niewiarowski, Density preserving homeomorphisms,Fund. Math.,106 (1980), 77–87. · Zbl 0447.28015 |

[4] | Krzysztof Ostaszewski, Continuity in the density topology,Real Anal. Exch.,7 (1981–82), 259–270. · Zbl 0494.26004 |

[5] | Krzystof Ostaszewski, The semigroup of density continuous functions,Real Anal. Exch. (to appear). |

[6] | A. Zygmund,Trigonometric Series, Vol. 1, Cambridge University Press, 1952. · JFM 58.0296.09 |

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