x ( t ) = x ( t, 0 , y (0)) of (2.1) which has an integral curve in D $^{r}$, and (2.5) holds. A solution x$_{0}$ ( t ) satisfies (2.1) and (2.4) simultaneously. Consider a solution x$_{0}$ ( t ) : $^{R}$→ R $^{n}$, x$_{0}$ ( t ) = x ( t, 0 , x$_{0}$ ) , x$_{0}$ ∈ D with discontinuity moments { θ$_{i}$ } . Fix a discontinuity moment θ$_{i}$. At this discontinuity moment, the trajectory may be on Γ and ˜ Γ . All possibilities of discontinuity moment should be analyzed. For this reason, we should investigate the following six cases: If a discontinuity point x$_{0}$ ( θ$_{i}$ ) satisfy the case ( α ) , (( α ′ )) the case ( β ) , (( β ′ )) and the case ( γ ) , (( γ ′ )) we will call it an ( α ) − type point, a ( β ) − type point and a ( γ ) − type point, respectively. Besides, we present the following definition which is compliant with Definition 2.2. Definition 2.5 If there exists a discontinuity moment, θ$_{i}$, i ∈ A , for which one of the cases ( γ ) or ( γ ′ ) is valid, then the solution x$_{0}$ ( t ) = x ( t, 0 , x$_{0}$ ) , x$_{0}$ ∈ R n of (2.1) is called a grazing solution and t = θ$_{i}$ is called a grazing moment. Next, we consider the differentiability properties of grazing solutions . The theory for the smoothness of discontinuous dynamical systems’ solutions without grazing phenomenon is provided in [1]. Denote by ¯ x ( t ) , j = 1 , 2 , . . . , n, a solution of (2.4) such that ¯ x (0) = x$_{0}$ + Δ x, Δ x = ( ξ$_{1}$, ξ$_{2}$, . . . , ξ$_{n}$ ) , and let η$_{i}$ be the moments of discontinuity of ¯ x ( t ) . The following conditions are required in what follows. where u$_{i}$ ( t ) ∈ P C ([0 , b ] , θ ) . ( C ) The discontinuity moment η$_{j}$ of the near solution approaches to the discontinuity moment θ$_{j}$, j ∈ A , of grazing one as ξ tends to zero. (B) There exist constants ν$_{ij}$, j ∈ A , such that (A) For all t ∈ [0 , b ] \ ∪ $^{i}$∈ A ̂ ( η$_{i}$, θ$_{i}$ ] , the following equality is satisfied ¯ x ( t ) − x$_{0}$ ( t ) = n ∑ i =1 u$_{i}$ ( t ) ξ$_{i}$ + O ( ‖ Δ x ‖ ) , (2.7) η$_{j}$ − θ$_{j}$ = n ∑ i =1 ν$_{ij}$ ξ$_{i}$ + O ( ‖ Δ x ‖ ); (2.8) ( α ) x$_{0}$ ( θ$_{i}$ ) ∈ Γ \ ∂ Γ , ( α ′ ) x$_{0}$ ( θ$_{i}$ ) ∈ ˜ Γ \ ∂ ˜ Γ , ( β ) x$_{0}$ ( θ$_{i}$ ) ∈ ∂ Γ & 〈∇ Φ( x$_{0}$ ( θ$_{i}$ )) , f ( x$_{0}$ ( θ$_{i}$ )) 〉 ̸ = 0 , ( β ′ ) x$_{0}$ ( θ$_{i}$ ) ∈ ∂ ˜ Γ & 〈∇ ˜ Φ( x$_{0}$ ( θ$_{i}$ )) , f ( x$_{0}$ ( θ$_{i}$ )) 〉 ̸ = 0 , ( γ ) x$_{0}$ ( θ$_{i}$ ) ∈ ∂ Γ & 〈∇ Φ( x$_{0}$ ( θ$_{i}$ )) , f ( x$_{0}$ ( θ$_{i}$ )) 〉 = 0 , ( γ ′ ) x$_{0}$ ( θ$_{i}$ ) ∈ ∂ ˜ Γ & 〈∇ ˜ Φ( x$_{0}$ ( θ$_{i}$ )) , f ( x$_{0}$ ( θ$_{i}$ )) 〉 = 0 . 8