I'm reading MWG's Microeconomics, and I'm a bit confused about the utility representation theorem. It states that a rational and continuous preference relation can be represented by a continuous utility function, but in the proof it gives, monotonicity is used as a condition.
Further, in the next chapter considering Walrasian correspondence, it says if the relation is locally nonsatiated, then the correspondence $x(p,w)$ satisfies Walras'law.
I am very confused here. So utility representation theorem only require rationality and continuity? But the proof even used strong monotonicity which is stronger than monotonicity, let alone locally nonsatiated.