A simple technique for calculation of numerical integration errors for physically meaningful functions is presented. In this paper, first the function classes are introduced that relate a function studied directly to a particular physical problem. The matrix elements defining a particular class of functions are then directly incorporated into the error formula. The proposed technique relies on the extension of the concept of the Taylor series. Since the standard Taylor series can be used for calculating the errors for smooth functions, in this paper the behaviour of the error in the immediate vicinity of discontinuity is considered only. The extended Taylor series is obtained by performing the Taylor series expansion on both sides of the discontinuity, using a matrix to describe the behaviour of the function at the discontinuity position and finally summing up all terms proportional to the function value and of all its derivatives. For illustration, several basic classes of physically meaningful functions are introduced and the extended Taylor series is derived. The series is then used to calculate the local error of the numerical integration. The numerical results obtained confirm the accuracy of the derived formulae.