Can Non Continuous Preferences Be Represented by a Continuous Utility Function
The implications of Completeness and the Continuity axiom for utility representation
Assumptions for existence of a utility function
In your question you imply that the only assumptions needed on preference to produce a a real-valued function (a utility function) that represents those preferences are completeness and transitivity. This is incorrect. To represent preferences with a real-value function, you need (1) completeness, (2) transitivity, (3) continuous preferences, and (4) local non-satiation. You can find a proof of this fact in Microeconomic Theory by Mas-Collel, Whinston, and Greene. A simpler proof using strict monotonicity in place of local non-satiation can be found in other books (Reny and Jehle).
The assumption of continuous preferences
To clarify, the axiom of continuity is the assumption that the preferences are continuous, not the resulting utility function. The preference relation $\preceq$ is continuous if, for any bundle $x \in X$, the set of all bundles at least as good as $x$, $$ \succeq(x) \equiv \{y \in X \mid y \succeq x \}, $$ is closed. In contrast to a continuous utility function, note that a set of preferences has infinitely many possible representations. For example, the preference relation on levels of money $m \in \mathcal R$ such that $m_1 \succeq m_2$ iff $m_1 \geq m_2$ is continuous as describe above. Now, notice that these two utility functions are both valid representations: $$ u_1(m) = m $$ and $$ u_2(m) = m + \boldsymbol 1\{m \geq 2\}. $$ These are both valid because $u_i(m_1) \geq u_i(m_2)$ iff $m_1 \geq m_2$, $i = 1,2$. Thus $u_i(m_1) \geq u_i(m_2)$ iff $m_1 \succeq m_2$, $i = 1,2$.
Example of Preferences that are not continuous
The classic example of preferences that cannot be represented with a real-valued function are lexicographic preferences. The idea behind the proof is that a utility function can represent the ordering along the first category in the lexicographic ordering, but afterwards there "are not enough numbers left" to represent the others. See Mas-Collel p. 46 for details.
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Comments
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Completenes means that every basket of goods in some set previously defined is comparable with the use of a complete preference. Now, with the additional assumption that the preferences are transitive, we can prove that this preference can be represented by an utility function. That sums up the importance of the axiom.
Normally continuity is defined as an additional assumption for utility functions in text-books of microeconomics, but why completeness does not imply continuity, or the latter, continuity, does not imply completeness?
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How do you find the minimum or maximum of a function that is not continuous? The usual method, using derivatives, breaks if the functions aren't continuous. (Consider a line decreasing with slope $-1$ and a single discontinuous step of height $+1$ somewhere. The function has a local maximum at the step, but has no global maximum...)
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But then again, by the definition of continuity you gave, why completeness does not guarantee that the set you mentioned will be closed (since any bundle in X is comparable)?
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Like I said, consider "Lexicographic preferences." For example, say there are two categories of goods, $x$ and $y$. I like $B_1 =(x_1, y_1)$ better than $B_2 = (x_2, y_2)$ when $x_1 > x_2$. When $x_1 = x_2$, I like $B_1$ more when $y_1 > y_2$. Certainly these preferences are complete. I can compare any bundle. However, they're not continuous. For all $n \geq 1$, $(2 + 1/n, 0) \succ (2,1)$. But $(2 + 1/n, 0) \xrightarrow{n} (2,0) \prec (2, 1)$. So the set $\succeq (2, 1)$ is not closed.
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Cool, no prob. Glad it helps. Also, if the answer looks ok, could you "accept" it?
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I would like to correct a statement in the otherwise excellent answer. It is not true that condition (4) local non-satiation is necessary for preferences to be representable by a utility function. While Mas-Collel, Whinston, Green assume monotonicity in their proof of the representation theorem, this is only for convenience. In fact, the classic representation theorem of Debreu is that a preference relation defined on a convex domain can be represented by a continuous utility function if the preference relation is (1) complete, (2) transitive, and (3) continuous.
Recents
What is the matrix and directed graph corresponding to the relation $\{(1, 1), (2, 2), (3, 3), (4, 4), (4, 3), (4, 1), (3, 2), (3, 1)\}$?
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Source: https://9to5science.com/the-implications-of-completeness-and-the-continuity-axiom-for-utility-representation
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