Some remarks on $ρ$-upper continuous functions

Rok, strany: 2010, 85 - 89

The notion of a $\rho$-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.: \textit{A note on~$\rho $-upper continuous functions}, Tatra. Mt. Math. Publ. \textbf{44} (2009), 153--158]. the authors proved that each $\rho$-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 $\rho$-upper continuous function for any $\rho\in[0,1)$ and there exists a function which is $\rho$-upper continuous for each $\rho \in [0,1)$ and is not approximately continuous. In the paper [Kowalczyk, S.---Nowakowska, K.: \textit{A~note on $\rho $-upper continuous functions}, Tatra. Mt. Math. Publ. \textbf{44} (2009), \hbox{153--158}] there is also proved that for each $\rho\in(0,\frac{1}{2})$ there exists a $\rho$-upper continuous function which is not in the first class of Baire. Here we show that there exists a function which is $\rho$-upper continuous for each $\rho\in[0,1)$ but is not Baire 1 function

ISO 690:

Karasińska, A., Wagner-Bojakowska, E. 2010. Some remarks on $ρ$-upper continuous functions. In Tatra Mountains Mathematical Publications, vol. 46, no.2, pp. 85-89. 1210-3195.

APA:

Karasińska, A., Wagner-Bojakowska, E. (2010). Some remarks on $ρ$-upper continuous functions. Tatra Mountains Mathematical Publications, 46(2), 85-89. 1210-3195.