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.
