The variable z appears in discrete (Z) transforms and the variable k (number of periods) can also appear

If we discretize our plant with the ZOH method the controller has to be discretized with the same method

The number of poles at one in discrete time play exactly the same role as the integrators in continuous time

Only the region of the left semiplane (s variable) from imaginary sampling frequency to minus imaginary sampling frequency and from -1 to 0 can be compressed and represented into the discrete unit circle (z variable)

The discrete Bode diagrams are drawn the same way as the continuous ones but their interpretation is different 

Diverse continuous signals can be obtained from a discrete time signal

If we discretize our plant with the ZOH method then its input will be step-like

If we discretize our plant with the ZOH method its continuous and discrete response to step will be the same period by period

Only a continuous system can replicate the operation of a discrete system

Once the sampling period is chosen and with a given resolution, only one discrete

time signal can be obtained from sampling a continuous signal