A Periodic Function
$ x(t) = sin(t) $
Clearly, $ x(t) $ is periodic because there is a $ T > 0 $ such that $ x(t + T) = x(t) $ for all $ t $. An obvious choice for $ T $ would be $ T = 2\pi $. Shifting $ x(t) $ by $ 2\pi $ gives the original function since $ 2\pi $ is the fundamental period of $ x(t) = sin(t) $
A Non-Periodic Function
$ x[n] = sin(n) $
At first glance, it would appear that $ x[n] $ is periodic following the same reasoning as above; however, because $ x[n] $ is a discrete function, this is not the case. The definition for a periodic discrete signal is that there exists an integer $ N > 0 $ such that $ x[n + N] = x[n] $ for all $ n $. The fundamental period of the sine function is $ 2\pi $, which is not an integer. Furthermore, any integer multiple of $ 2\pi $ is not an integer. Therefore, $ x[n] = sin(n) $ is not periodic.
