I have a question when I'm doing exercise 3.C.5(b) of MWG. The exercise asks to prove that a continuous preference on $(-\infty,\infty)\times R^{L-1}_+$ is quasilinear with respect to the first commodity iff it admits a utility function $u(x)$ of the form: \begin{equation} u(x)=x_1+\Phi(x_2,...,x_L) \end{equation} In the proof, the existence of the utility function needs to be shown, but I am not sure if the quasilinear preference is monotonic (and rational) since the definition of quasilinear preference only states about the numeraire commodity.
My question: is the quasilinear preference relation naturally rational, monotonic and continuous, or we have to state the rationality, monotonicity and continuity additionally? Could you give some references?
Thanks a lot!
