On the Marcinkiewicz theorem for the binary Perron integral

A criterion for Perron integrability of the derivative of a function is stated in terms of the variation of the function with respect to a differential base. The criterion is used to construct an example showing that the Marcinkiewicz theorem, which asserts that ordinary Perron integrability follows from the existence of at least one continuous Perron minorant and at least one continuous Perron majorant, cannot be generalized to the binary Perron integral.