If $f$ is a function with domain $D\subset \mathbb R$, the domain of its derivative $f'$ is a subset of $D$. This leads to the definition of the descending sequence of sets:
$$ D= D_0 \supset D_1 \supset D_2 \supset D_3 \supset \cdots \supset D_n \supset D_{n+1} \supset \cdots$$
where $D_k$ is the domain of $f^{(k)}$. We can then define the limit $D_\infty$ of the sequence $(D_n)_{n\ge 0}$ as
$$ D_\infty = \lim_{n\to\infty} D_n = \bigcap_{n\ge 0} D_n $$
What does $D_\infty$ look like?
In particular,
- Does the sequence stabilize (become constant after some rank)?
- What are the topological properties of $D_\infty$?
- Can any subset of $\mathbb R$ be the $D_\infty$ of some function $f$?
I tried some simple example, but I lack imagination.