RO-NMO 2004 Problem 114

(a) Build a function $f : \mathbb R \to \mathbb R_+$ with the property $\left( \mathcal P \right)$ , i.e. all $x \in \mathbb Q$ are local, strict minimum points.

(b) Build a function $f : \mathbb Q \to \mathbb R_+$ such that every point is a local, strict minimum point and such that $f$ is unbounded on $I \cap \mathbb Q$ , where $I$ is a non-degenerate interval.

(c) Let $f: \mathbb R \to \mathbb R_+$ be a function unbounded on every $I \cap \mathbb Q$ , where $I$ is a non-degenerate interval. Prove that $f$ doesn't have the property $\left( \mathcal P \right)$ .