### 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

- mr: 3016317
- zbl: 1259.28005
- doi: 10.2478/s11533-012-0144-1

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.