While the z-transform method is a basic mathematical tool to relate the imput/output signals only at the sampling instants in analyzing and designing sampled-data control systems, the modified z-transform which is a variation of the z-transform is widely used to represent the details of continuous signals between the sampling instants. To relate the modified z-transform to the corresponding regular z-transform, some properties were established regarding the modified z-transform method. This paper will show that these properties, in their current forms, cause come analytic problems, when they are applied to the signals with discontinuities at the sampling instants, which accordingly limit their applications significantly. In this paper, those analytic problems will be investigated, and the theorems of the modified z-transform will be revised by adopting new notations on the z-transform so that those can be correctly interpreted and used without any analytic problems. Also some additional useful schemes of applying the modified z-transform will be developed.