Call a function $f: \mathbb R+ \to \mathbb R$ **essentially $C^\infty$** if there exists a sequence $f_n$ $(n \geq 0)$ such that each $f_n$ is differentiable a.e., $f_0 = f$ a.e., and $f_n’$ is equal to $f_{n+1}$ a.e.

Given a essentially $C^\infty$ function $f$, and two associated sequences $f_n$ and $g_n$, it is true that necessarily $f_n = g_n$ a.e - so to every essentially $C^\infty$ function $f$, we can assign a unique sequence of equivalence classes $[f_n]$.

Call a essentially $C^\infty$ function h **essentially anti-Cauchy** if for every $M > 0$ there exists some $N > 0$ such that $|h’| > M$ a.e. on $(N, \infty]$.

For every subset $S$ of the naturals, does there exist a essentially $C^\infty$ function $f$ such that $f_n$ is essentially anti-Cauchy exactly when $n$ is in $S$?