Periodic
A function is period if there is a time interval $ T > 0\! $, such that the value for the function at time $ t\! $ is equal to the value of the function at time $ t + T\! $. An example of this would be:
$ f(x) = cos(x)\! $
Note that this is only a periodic function when it comes to the continuous time realm. In the discrete world, cosine is surprisingly not periodic, but that is another story for another time (discrete time, to be exact).
The proof for the continuity of this function is rather simple. Suppose $ T = 2\pi\! $. If this is the case, and cosine is continuous as I claimed, then cosine of $ t = 2\! $, $ t = 4\! $, and so forth would all be equal. If you test this, it is, in fact, the case. Ha, I hate to say I told you so, but I told you so.
But wait! Can't $ T\! $ also be $ 4\pi\! $, $ 8\pi\! $, $ 10,000\pi\! $, or any multiple of $ 2\pi\! $? The answer is yes! However, $ 2\pi\! $ is the smallest $ T\! $ we can have, which we will refer to as the fundamental period. Every other value of $ T\! $ is just another period.
Non-periodic
A non-periodic function is one that does not have a value for $ T\! $ that satisfies the conditions given for the periodic function. If you couldn't conclude that from the name "non-periodic", you aren't very bright. However, if you want an example of such a function, here you go:
$ f(x) = e^x\! $
I bet there is no $ T\! $ that makes this periodic. If you can find one, I will give you a million bucks.
