where the matrix A ( t ) ∈ R $^{n}$×n of the form A ( t ) = ∂f ( x$_{0}$ ( t )) ∂x . The matrices B$_{i}$, i = 1 , . . . , n, will be defined in the remaining part of the paper. The matrix B$_{i}$ is bivalued if θ$_{i}$ is a grazing moment or of ( β ) − type. The right hand side of the second equation in (3.9) will be described in the remaining part of the paper for each type of the points. As the linearization at a point of discontinuity , we comprehend the second equation in (3.9). Discontinuity points of ( α ) and ( α ′ ) types are discussed in [1]. In this subsection, we will outline the results of the book. Assume that x ( θ$_{i}$ ) is an ( α ) − type point. It is clear that the B − equivalent system (2.4) can be applied in the analysis. The functions τ$_{i}$ ( x ) and W$_{i}$ ( x ) , are described in Subsection 2.1. Differentiating Φ( x ( τ$_{i}$ ( x ))) = 0 , we have Then, considering (2.3), we get the following equation, where e$_{j}$ = (0 , . . . , 1 ︸ ︷︷ ︸ j , . . . , 0) . The matrix B$_{i}$ ∈ R $^{n}$×n The matrix B$_{i}$ ∈ in equation (3.9) is defined as B$_{i}$ = W$_{ix}$, where W$_{ix}$ is the n × n matrix of the form W$_{ix}$ = [ ∂W$_{i}$ ( x$_{0}$ ( θ$_{i}$ )) ∂x$_{1}$ , ∂W$_{i}$ ( x$_{0}$ ( θ$_{i}$ )) ∂x$_{2}$ , . . . , ∂W$_{i}$ ( x$_{0}$ ( θ$_{i}$ )) ∂x$_{n}$ ] . Its vector-components ∂W$_{i}$ ( x$_{0}$ ( θ$_{i}$ )) ∂x$_{0}$$_{j}$ , j = 1 , . . . , n, evaluated by (3.11). Moreover, the components of the gradient ∇ τ$_{i}$ have to be evaluated by formula (3.10). In what follows, denote n × n zero matrix by O$_{n}$. In the light of the possibilities ( N 1) and ( N 2) , the matrix B$_{i}$ in (2.1) can be expressed as follows: 3.2 Linearization at ( β ) − type points where W$_{ix}$ is evaluated by formula (3.11) and ∇ τ ( x ) evaluated by formula (3.10). The differentiability properties for the cases ( α ′ ) and ( β ′ ) can be investigated similarly. B$_{i}$ =    $^{O$_{n}$}$$^{,}$ if ( N 1) is valid, W$_{ix}$, if ( N 2) is valid, (3.12) 10 3.1 Linearization at ( α ) − type points ∂τ$_{i}$ ( x$_{0}$ ( θ$_{i}$ )) ∂x$_{j}$ = − Φ$_{x}$( x$_{0}$ ( θ$_{i}$ )) ∂x$_{0}$ ( θ$_{i}$ ) ∂x$_{0}$$_{j}$ Φ$_{x}$( x$_{0}$ ( θ$_{i}$ )) f ( x$_{0}$ ( θ$_{i}$ )) . (3.10) ∂W$_{i}$ ( x$_{0}$ ( θ$_{i}$ )) ∂x$_{0}$$_{j}$ = ( f ( x$_{0}$ ( θ$_{i}$ )) − f ( x$_{0}$ ( θ$_{i}$ ) + J ( x$_{0}$ ( θ$_{i}$ )))) ∂τ$_{i}$ ∂x$_{0}$$_{j}$ + ∂I ∂x ( e$_{j}$ + f ∂τ$_{i}$ ∂x$_{0}$$_{j}$ ) , (3.11)