Is only fluid velocity or both fluid velocity and sound speed discontinuous at a shock?

In any shock wave, there will be a discontinuous change in the bulk flow velocity and relevant communication speed. In a neutral fluid like the Earth's atmosphere, the communication speed is the speed of sound. In the ionized gas of space, called a plasma, there are many relevant communication speeds (e.g., see https://physics.stackexchange.com/a/179057/59023).

The speed of sound is defined as: $$ C_{s}^{2} = frac{ partial P }{ partial rho } tag{0} $$ where $P$ is the total thermal pressure and $rho$ is the mass density of the fluid. The Rankine–Hugoniot relations show us that change in $rho$ (discussed at https://physics.stackexchange.com/a/349724/59023) can be written as: $$ frac{ rho_{dn} }{ rho_{up} } = frac{ left( gamma + 1 right) M_{up}^{2} }{ left( gamma + 1 right) + left( gamma - 1 right) left( M_{up}^{2} - 1 right) } = frac{ U_{up} }{ U_{dn} } tag{1} $$ where $rho_{j}$ is the mass density in region $j$, $M_{j}$ is the Mach number in region $j$, $U_{j}$ is the bulk fluid flow speed along the shock normal unit vector in region $j$, where we use the subscripts $up$ and $dn$ for upstream(pre-shock) and downstream(shocked), respectively. The change in pressure (discussed at https://physics.stackexchange.com/a/523114/59023) can be written as: $$ frac{ P_{dn} }{ P_{up} } = frac{ 2 gamma }{ gamma + 1 } M_{up}^{2} - frac{ gamma - 1 }{ gamma + 1 } tag{2} $$

In a hydrodynamic shock, the change in $rho$ is limited to 4 for polytropic index $gamma$ = 5/3 (i.e., take the limit as Mach number goes to infinity in Equation 1). The change in pressure, in contrast, is not limited and can diverge just as the Mach number can (in principle this is true, but in reality the Mach number can only get really large, not infinite).

I am having the doubt whether only $v(x)$ or both $v(x)$ and $C_{s}$ have a discontinuity at the shock.

As you can see from Equations 1 and 2, the pressure and mass density do not change by the same amount across hydrodynamic shocks. Thus, $C_{s}$ will differ on either side of the shock, as shown by Equation 0.