<abstract xmlns="http://www.w3.org/1999/xhtml"><p> The notion of a <italic>ρ</italic>-upper continuous function is a generalization of the notion of an approximately continuous function. It was introduced by S. Kowalczyk and K. Nowakowska. In [Kowalczyk, S., Nowakowska, K.: <italic>A note</italic><italic>on ρ-upper continuous functions</italic>, Tatra. Mt. Math. Publ. <bold>44 </bold>(2009), 153-158]. the authors proved that each <italic>ρ</italic>-upper continuous function is measurable and has Denjoy property. In this note we prove that there exists a measurable function having Denjoy property which is not <italic>ρ</italic>-upper continuous function for any <italic>ρ </italic>∈ [0<italic>, </italic>1) and there exists a function which is <italic>ρ</italic>-upper continuous for each <italic>ρ ∈ </italic>[0<italic>, </italic>1) and is not approximately continuous. In the paper [Kowalczyk, S.-Nowakowska, K.: <italic>A note on ρ-upper continuous functions</italic>, Tatra. Mt. Math. Publ. <bold>44 </bold>(2009), 153-158] there is also proved that for each <italic>ρ </italic>∈<italic></italic>(0<italic>, </italic>1 2 ) there exists a <italic>ρ</italic>-upper continuous function which is not in the first class of Baire. Here we show that there exists a function which is <italic>ρ</italic>-upper continuous for each <italic>ρ ∈ </italic>[0<italic>, </italic>1) but is not Baire 1 function.</p></abstract>

The notion of a ρ-upper continuous function is a generalization of the notion of an approximately continuous function. It was introduced by S. Kowalczyk and K. Nowakowska. In [Kowalczyk, S., Nowakowska, K.: A noteon ρ-upper continuous functions, Tatra. Mt. Math. Publ. 44 (2009), 153-158]. the authors proved that each ρ-upper continuous function is measurable and has Denjoy property. In this note we prove that there exists a measurable function having Denjoy property which is not ρ-upper continuous function for any ρ ∈ [0, 1) and there exists a function which is ρ-upper continuous for each ρ ∈ [0, 1) and is not approximately continuous. In the paper [Kowalczyk, S.-Nowakowska, K.: A note on ρ-upper continuous functions, Tatra. Mt. Math. Publ. 44 (2009), 153-158] there is also proved that for each ρ ∈(0, 1 2 ) there exists a ρ-upper continuous function which is not in the first class of Baire. Here we show that there exists a function which is ρ-upper continuous for each ρ ∈ [0, 1) but is not Baire 1 function.

Some remarks on ρ-upper continuous functions

University of Ł´od´z Faculty of Mathematics and Computer Science ul. Banacha 22 PL–90-238 Ł´od´z POLAND

Tatra Mountains Mathematical Publications

{"article-title":"Some remarks on ρ-upper continuous functions"}

The notion of a ρ-upper continuous function is a generalization of the notion of an approximately continuous function. It was introduced by S. Kowalczyk and K....

Some remarks on ρ-upper continuous functions

Published Online: Nov 13, 2012

Page range: 85 - 89

DOI: https://doi.org/10.2478/v10127-010-0023-8

Keywords
density of a set at a point, ρ-upper continuous functions, approximately continuous functions, Baire 1 functions, Denjoy property.

This content is open access.

Some remarks on ρ-upper continuous functions

Published Online: Nov 13, 2012

Page range: 85 - 89

DOI: https://doi.org/10.2478/v10127-010-0023-8

Keywordsdensity of a set at a point, ρ-upper continuous functions, approximately continuous functions, Baire 1 functions, Denjoy property.

This content is open access.

Keywords
density of a set at a point, ρ-upper continuous functions, approximately continuous functions, Baire 1 functions, Denjoy property.