Assume that the set of values where $f^{(k)}≠0$ is finite

Question

Let $f:ℝ→ℝ$ be a real analytic function. Let $f^{(k)}$ be the $k$th derivative of $f$. Assume that the set of values where $f^{(k)}≠0$ is finite, then what we can say about the function $f$.

Answer 1

The basic idea is that derivatives have the intermediate value property. That is, if $g$ is differentiable on an open interval $I$ and $g'(a)<\alpha<g'(c)$ for some $a,c\in I$ then there exists $b\in(a,c)$ such that $g'(b)=\alpha$. So if $g'$ takes only finitely many values on $I$, this is pretty strong information which should enable you to find a simple description of $g$.

Then to deal with $f^{(k)}$, just take $g=f^{(k-1)}$ in the previous paragraph to get a description of $f^{(k-1)}$. Then integrate repeatedly to find $f$. (And don't worry, $f^{(k-1)}$ will be integrable.)