The set of polynomials of degree $n-1$ is the pre-image of the constant zero function ($\bar{0}(x)=0$) with respect to the $n$th derivative. What is the analogous pre-image of a neighborhood of $\bar{0}$, and how does it compare to the general space of functions under consideration?
More concretely: What metrics might be natural to consider here, and within what class of functions? What properties would one consider interesting to seek within such a set, and what pre-existing theories might be useful to tackle such a question?
