# Show that there is no continuous utility function which represents the lexicographic preferences [duplicate]

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 $x_2 > y_2$.

• 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. – Kitsune Cavalry Nov 1 '15 at 16:01
• 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 $x_2<y_2$. – Herr K. Nov 1 '15 at 18:42

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