Do derivatives make sense for such a function. I.e. suppose we have a real valued function $f:A\to\mathbb{R}$, where $A=\{1,2,3,4,5\}$.
Question Is there a notion of derivative for such a function? If it is not the standard notion, why does the standard notion not apply?
I realize this may be a dumb question, but for derivatives I've always worked with the domain being $\mathbb{R}$ and have not thought about whether derivatives and other calculus techniques apply to other domains.
(so a more general question if someone wants to tackle it), is what requirements on a functions domain/range are required for standard calculus ideas to apply
I tried googling a bit, and come across something called "finite differences" and "finite calculus" which maybe deal with what I am asking.
There is a notion of "discrete derivative" which is useful when dealing with discrete\finite groups.
If $G$ is a group and $f:G\rightarrow\mathbb{R}$ a function. We can define a "discrete derivative in the direction $h\in G$" by the formula $\Delta_h f(g)=f(h+g)-f(g)$.
In particular this makes sense for finite groups like $G=\mathbb{Z}/k\mathbb{Z}=\{0,1,2,...,k\}$. Also you can use these derivatives to define the notion of polynomials from $G$ to $\mathbb{R}$. i.e. We say that a function $f:G\rightarrow\mathbb{R}$ is a polynomial of degree $k$ if any for every $h_1,...,h_k\in G$ we have that $\Delta_{h_1}...\Delta_{h_k} f = 0$
Note also that this operation really acts like a derivative. For example, if $\Delta_h f = 0$ for every $h\in G$ then $f$ is a constant. More generally, if $\Delta_h f = \Delta_h g$ for all $h\in G$ then $f,g$ differ by a constant (so an anti-derivative is only unique up to constant).