For the top (continuous) branch, keep in mind that the Derivative block is a numerical approximation and it's output will be very dependent on the solver parameters. I was able to get the expected result by using a Fixed Step solver (ode2) with a Fixed step size of 0.002. Don't forget to the set the Initial condition of Integrator4 to 1. In general, using Derivative block is not recommended and using two in series with each other will likely be very problematic.
For the bottom (discrete) branch, change the Integrator method on both Discrete Time Integrators to Backward Euler. Place a Zero Order Hold block between the Sine Wave and the first Discrete Derivative and set its Sample time to what you want (I used 0.1). Then adjust the Intital Condition parameters on each Discrete Derivative and Discrete Time Integrator so that the output of each block has the desired value at T=0.