# What are all the elements of $\mathrm{Fun}(\mathbb{Z},\mathbb{Z})$ that satisfy the following property?

I'm very certain that this is nothing new under the sun but this is something I wondered yesterday and couldn't find resources online. Let $$R$$ be a (commutative, unital) ring and let $$I$$ be an ideal in $$R$$. I say that a function $$f:R\rightarrow R$$ is $$I$$-differentiable if for all $$x\in R$$ we have $$f(x+I)-f(x)\subset I$$. The set $$D(I)$$ of all such functions is easily seen to be a subring of the ring $$\mathrm{Fun}(R,R)$$ of all $$R$$-valued functions on $$R$$ that contains all polynomial functions, all $$I$$-invariant functions and is stable und composition. On the other hand, in many cases such as $$R=\mathbb{Z}$$ and $$I=n\mathbb{Z}$$, $$D(I)$$ is easily seen to be not the whole ring (e.g. no nontrivial function with finite support is $$n\mathbb{Z}$$-differentiable). So my question is: What precisely are all the $$n\mathbb{Z}$$-differentiable $$\mathbb{Z}$$-valued functions on $$\mathbb{Z}$$? I'm suspecting $$D(n\mathbb{Z})$$ to actually be the smallest subring that contains all polynomial functions, all $$I$$-invariant functions and is stable under composition, but would be delighted if it was actually larger.

Thank you so much for your help and for providing further literature!

• The $I$-differentiable functions are exactly the ones which induce a well-defined function $R/I \to R/I$ Sep 6, 2023 at 20:08
• So e.g. for $I = n \mathbb{Z}$, you can think of an $n \mathbb{Z}$-differentiable function just as a collection of $n$ endofunctions of $\mathbb{Z}$ Sep 6, 2023 at 20:22
• Ah, okay, that makes sense; thank you! Sep 7, 2023 at 7:40