||In this thesis, we obtained some generaliszations of a theorem of Marcinkiewicz. The main tools of the proofs in this thesis are Weierstrass' approximation theorem, Lusin's theorem and Egoroff's theorem. In the one variable generalization, we will see that the statement holds even when we replace the closed and bounded interval I by R, the real line. We will also prove the statement with the "first difference form" replaced by the "second difference form". In the two variables generalization, we will first construct a "Cantor function" on [0,1] x [0,1] . With this function in hand, we can prove on R2 a similar statement with some variations. Finally we will see some linkages between the one variable generalization and the two variable generalization.