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Maximums of upper semicontinuous strong Świątkowski functions

100%

Szczuka P.

tom Vol. 44, nr 1

59-65

In this paper we characterize the family of the maximums of upper semicontinuous strong Świątkowski functions and we show that the family of all upper semicontinuous strong Świątkowski functions is not closed with respect to maximums.

Quotients of quasi-continuous functions

86%

Jałocha J.

2000

tom Vol. 6, nr 2

251-258

The main goal of this paper is to characterize both the quotients of quasi-continuous and the quotients of Darboux quasi-continuous functions. We prove also theorems concerning common divisor for the families of the quotients of quasi-continuous (Darboux quasi-continuous) functions with respect to quasi-continuity (Darboux property and quasi-continuity, respectively).

On some properties of Hamel bases and their applications to Marczewski measurable functions

58%

Dorais F. , Filipów R. , Natkaniec T.

tom 11

nr 3

487-508

We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.