This is a 2010 Tier-1 (Qualifying exam for Ph.D. students at IUB) Analysis problem. **Question** A function $f:\mathbb{R}\mapsto\mathbb{R}$ is //proper// if $f^{-1}(C)$ is compact for any compact set $C$. Suppose that $f$ and $g$ are both continuous and proper. Prove or give a counterexample: Is $fg$ proper?