Does there exist such an equation that the differences, are infinite? And if there isn't then what is the proof?
For example:
$x^2$ has a finite difference of 2.
$ \begin{array}{c|c|c|c} n & y & \text{first differences} & \text{second differences} \\ \hline 1 & 1 & 0 & 0 \\ 2 & 4 & 3 & 2 \\ 3 & 9 & 5 & 2 \\ 4 & 16& 7 & 2 \\ 5 & 25& 9 & 2 \\ \end{array} $
Therefore, all of the second differences are the same. Furthermore, this found by subtracting the first differences e.g. $5-3 = 2$ and $7-5=2$ etc.