Example of a Periodic Function is:
e^(1/2)*j$ pi $n
This is discrete sigal.
Therefore (w/2*pi) should be rational.
In this above example the value is (w/2*pi) is 1/4.
Hence it is a periodic function.
Example of a non-periodic function:
cos(n)
This is a discrete signal too.
Therefore the criteria for it to be peridic is cos(n+N)=cos(n).
Where "N" is any integer.
In the above example N is 2*pi.
Though it looks like it is periodic, N is not an integer.
Therefore cos(n) is not a periodic function.
