The finite-time stabilization and suppression of matched disturbances using discontinuous controls with limited magnitudes is the main advantage of automatic control systems operating in sliding mode. However, these methods are not always implementable in actual plants. Frequently, authors in their papers provide an excellent theoretical background for sliding mode control. Then, they present the results of practical experiments in which discontinuous controls were replaced by continuous analogs (but without any mathematical proof). However, such systems with continuous controls do not fully correspond to an a priori theoretical description. In this paper, this gap is filled, and a rigorous analytical analysis of this replacement is performed using the example of a fully actuated vector system with an uncertain input matrix. First, we describe methods for synthesizing discontinuous controls, including the hierarchy method. Then, we formalize the properties of the hyperbolic tangent. This function can be considered as a continuous smooth analog of the sign function. Next, we develop methods for synthesizing the corresponding control and its justification. The obtained results eliminate the aforementioned discrepancies between the theoretical and practical realizations. To illustrate the developed algorithms, we present the results of numerical modeling of the control system of a manipulator endpoint with three degrees of freedom functioning under uncertainty are presented.