The ring $R$ of all functions from $\mathbb{R}$ to $\mathbb{R}$ may be identified with the infinite Cartesian product $\prod_{x \in \mathbb{R}} \mathbb{R}$. This suggests consideration of properties of an arbitrary Cartesian product $R = \prod_{i \in I} R_i$ of commutative rings: that is, $I$ is some index set and for each $i \in I$, $R_i$ is a commutative ring. In this level of generality, it is straightforward to show:
1) An element $x \in R$ is a zero divisor iff at least one of its coordinates $x_i$ is a zero divisor in $R_i$.
2) An element $x \in R$ is nilpotent iff there exists $N \in \mathbb{Z}^+$ such that $x_i^N = 0$ for all $i \in I$. In particular, every coordinate $x_i$ of a nilpotent element is nilpotent.
3) An element $x \in R$ is a unit iff $x_i$ is a unit in $R_i$ for all $i \in I$.
In the case where each $R_i$ is a field, these observations imply that there are no nonzero nilpotent elements, and also: an element $x$ is a unit iff $x_i \neq 0$ for all $i \in I$; otherwise $x$ is a zero divisor.