CT periodic signal
An example of a periodic signal in continuous time is:
$ f(x) = cos(x)\! $
The period of the signal in the continuous time domain is $ 2\pi\! $. However, in discrete time, the same signal may or may not be periodic, depending on the rate, at which one samples the signal.
DT non-periodic signal
If the same signal, $ f(x) = cos(x)\! $, was sampled at the rate of one unit, the signal would not be periodic in the discrete world, as it will not fit the definition of a periodic signal. Basically, there is no number $ T\! $ that can be added to any time to create the same value.
DT periodic signal
However, the same signal, when sampled at a different rate, can be periodic in the discrete world. Suppose you sample the signal at the rate $ 2\pi\! $, $ \pi\! $, or $ 0.5\pi\! $. You will find that the discrete signal that is a result of the sampling is periodic, and it's period is the same as its period in the discrete world is, in fact, the same as its period in the continuous world. In fact, any multiple or factor of the signal's period in continuous time can be used as a sampling rate that will yield a periodic function in the discrete domain.
