Let D $\subset \mathbb{R^n}$ be an open convex domain and let f : D $\rightarrow \mathbb{R}$ be a map such that f has a locally strict maximum and a locally strict minimum.
Prove: The function f is neither quasi-convex nor quasi-concave.
I am trying to prove this property but my textbook gives very little information about quasi-convexity (quasi-concavity) at all to come up with an intelligent proof, so I have no idea where to start (or which properties to use).
If $p$ is a locally strict maximum with $f(p) = r$, then for $\epsilon > 0$ sufficiently small $\{x: f(x) \le r-\epsilon\}$ contains a sphere around $p$ but not $p$ itself, so is not convex, and $f$ is not quasi-convex. Similarly...
EDIT: