Notion of Derivative in Discrete Space We are accustomed with the notion of Calculus in $\mathbb{R}$. Can we define the notion of derivative/non-derivative in $\mathbb{N}$. I was thinking something as follows: Let $f$ be a function from $\mathbb{N}$ to itself. The function $f$ is non-differentiable at a point k if and only if $f(k-1)\neq f(k)\neq f(k+1)$. What do you people think? Does the notion of non-differentiability make sense?

There is a mathematics of finite differences, sometimes called the calculus of finite differences: <

As you can see, it has a number of notable analogues to the (more familiar) calculus of a real variable.