IMC 2009 · Day 1 · P1

a) No. Counter-example: $f$ and $g$ can be chosen as the characteristic functions of $[\sqrt{3}, \infty)$ and $(\sqrt{3}, \infty)$, respectively.

b) Yes. By the assumptions $g - f$ is continuous on the whole real line and nonnegative on the rationals. Since any real number can be obtained as a limit of rational numbers we get that $g - f$ is nonnegative on the whole real line.