Question: Suppose that \(\succeq\) is a rational preference relation on \(R_{+}^{L}\).

1. State the definition of a utility function representing \(\succeq\).
2. Is there always a utility function representing \(\succeq ?\) Why or why not?
3. If \(\succeq\) on \(R_{+}^{L}\) is also continuous, is there always a utility function representing \(\succeq\) ?
4. Suppose there is a continuous utility function representing \(\succeq\). Are all the utility functions representing \(\succeq\) continuous? Why or why not?

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