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Show that there is no continuous utility function which represents the lexicographic preferences $\mathscr{L}$ given by $(x_1, x_2) \succeq (y_1, y_2)$ if and only if $x_1 > y_1$ or $x_1 = y_1$ and $x_2 > y_2$.

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  • $\begingroup$ Hi, and welcome to Economics SE. If you could try to tell us what part of this proof you are having trouble with, it would allow us to help you better. That, and we don't usually allow flat out homework questions without any effort shown. $\endgroup$ – Kitsune Cavalry Nov 1 '15 at 16:01
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    $\begingroup$ Hint: Suppose there is such a utility function. Try to find a contradiction with the bundles $(x_1,x_2)$ and $(y_1,y_2)$ where $x_1>y_1$ but $x_2<y_2$. $\endgroup$ – Herr K. Nov 1 '15 at 18:42
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Hint:

To ensure the existence of utility function representation, we must have continuous preferences (and this is a sufficient condition). From the Mas-Colell book:

The preference relation $\succeq$ on $X$ is continuous if it is preserved under limits. That is, for any sequence of pairs ${(x^n, y^n)}^\infty_{n=1}$ with $x^n \succeq y^n$ for all $n$, $x = \lim_{n \to \infty} x^n$ and $y = \lim_{n \to \infty} y^n$, we have $x \succeq y$.

We then look at the definition of utility function representation.

The function $U$ represents $\succeq$ on $X$ if $\forall \ x,y \in X, x \succeq y \iff u(x) \geq u(y)$, where $u: X \to \mathbb{R}$.

Though it may be easier just to show that the preferences themselves are not continuous, and thus there is no utility function representation, continuous or not.

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