SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension

We prove the SBV regularity of the characteristic speed of the scalar hyperbolic conservation law and SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux function for general hyperbolic systems of conservation laws. More precisely, for the equation
$$
u_t + f(u)_x = 0, \quad u : \mathbb{R}^+ \times \mathbb{R} \to \Omega \subset \mathbb{R}^N,
$$
we only assume that the flux $f$ is a $C^2$ function in the scalar case ($N=1$) and Jacobian matrix $Df$ has distinct real eigenvalues in the system case $(N\geq 2)$. We show that for the scalar equation $f'(u)$ belongs to the SBV space, and for system of conservation laws the $i$-th component of $D_x\lambda_i(u)$
has no Cantor part, where $\lambda_i$ is the $i$-th eigenvalue of the matrix $Df$.