The part about derivatives might have seemed a little odd. After all, you might think, Z is a discrete set, so what does it mean to take derivatives of functions on it. One answer to this is to just differentiate symbolically using polynomial differentiation rules. But I think a better answer is to remember that we’re using a different metric than usual, and Z isn’t discrete at all! Indeed, for any number k, limn→∞k+pn=k, so no points are isolated, and we can define differentiation of functions on Z in exactly the usual way with limits.
The part about derivatives might have seemed a little odd. After all, you might think, Z is a discrete set, so what does it mean to take derivatives of functions on it. One answer to this is to just differentiate symbolically using polynomial differentiation rules. But I think a better answer is to remember that we’re using a different metric than usual, and Z isn’t discrete at all! Indeed, for any number k, limn→∞k+pn=k, so no points are isolated, and we can define differentiation of functions on Z in exactly the usual way with limits.