Define the rotation number $\rho(f)$ of an orientation-preserving circle map $f$ and the rotation number $\rho(F)$ of a lift $F$ of $f$. Prove that $\rho(f)$ and $\rho(F)$ are well-defined. Prove also that $\rho(F)$ is a continuous function of $F$.

State without proof the main consequence of $\rho(f)$ being rational.