[JadeNB]

> But an important detail is that the operator that is discussed in this post is not linear, i.e. D(x+y) != D(x) + D(y), so it doesn't have all the expected ("intuitive") properties that the usual derivative has.

I think some care is appropriate here. The property you quote is additivity, not linearity; for linearity, one would like a ground field (or a ground ring if one is discussing linear maps between modules, I suppose). Since one is not considering any vector space / module structure on the natural numbers, this may hint why linearity (or even its weaker sibling additivity) is not thought to be necessary here.

(EDIT: With that said, I like very much your description of transforming numbers to functions.)