Periodic Functions
A Continuous Time signal is said to be periodic if there exists $ T > 0 $ such that $ x(t+T)=x(t) $
A Discrete Time signal is said to be periodic if there exists $ N > 0 $ (where N is an integer) such that $ x[n+N]=x[n] $
An example of a DT periodic signal is $ x[n] = j^{n} $ because
$ x[1] = j $
$ x[2] = -1 $
$ x[3] = -j $
$ x[4] = 1 $
$ x[5] = j $
$ x[6] = -1 $
$ x[7] = -j $
$ x[8] = 1 $
As you can see the function has a fundamental period of 4. Therefore any multiple of 4 is a period.
Non-periodic Functions
A Continuous Time signal is said to be non-periodic if there is no value of $ T > 0 $ that satisfies $ x(t+T)=x(t) $
A Discrete Time signal is said to be non-periodic if there is no value of $ N > 0 $ (where N is an integer) that satisfies$ x[n+N]=x[n] $
An example of a non-periodic continuous time signal would be $ \ x(t) = e^{(-1 + j)t} $. This goes to show that not all complex exponential functions are periodic.
Here is what the function looks like when graphed:
