On some properties of Hamel bases and their applications to Marczewski measurable functions
By F. G. Dorais, R. Filipów and T. Natkaniec
Central European Journal of Mathematics 11 (2013), no. 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.