A solution to the problem on describing experimentally obtained dependences is considered. The autho r’s method is based upon getting some local approximations of fragments of these relations, and their additive reduction to a single analytical expression. This effect is determined using special “allocating” functions limitin g the domain of non zero definition for each of the approximation functions. The method is called “cut - glue” according to the applied principles. The closest analogue of the proposed method is spline approximation. However, the “cut - glue” method is much more adaptable, as it is bonded to neither the number of spline - approximable points, nor the function order approximating the areas. The order of the polynomial approximant, or another approximating function, as well as its structure for each site, can be arbitrary. Another advantageous difference of “cut - glue” approximation consists in a single a nalytic notation of the whole piecewise function instead of defining a vector spline - function through a cu mbersome system of equations. This effect has been achieved using an analytical function approximatin g and par ametrically arbitrarily approaching the Heaviside step function. The analytical and numerical studies of the properties and the effects of applying the proposed method are resulted. The obtained results are illustrated w ith the specific technical sample applications of the method to practical pr oblems, tabular and graphical data.