The article discusses method for approximating a three-dimensional body, which is described by a discontinuous 3D function. The case is considered when the body under study is completely covered by a system of elementary rectangular elements (parallelepipeds). As a discontinuous structure, the article constructs and studies a discontinuous spline interpolant, which uses one-sided values of the function under study in a given rectangular grid of nodes to construct. Theorems on the interpolation properties and errors of the constructed discontinuous structures are formulated. Moreover, the constructed discontinuous interpolation splines include, as a special case, classical continuous interpolation splines. The article tests the hypothesis that discontinuous functions should also be approximated by discontinuous construction. Such approximation methods can be used in remote methods for studying objects, namely, in computer and seismic tomography, in digital signal processing, in non-destructive quality control of products, etc. In the future, the authors plan, based on the created methods, to approximate discontinuous functions, determine methods for determining points, lines and planes of discontinuity, and then apply these methods for mathematical modeling in remote sensing methods.