Can Continuous Preferences Be Represented by a Discontinuous Utility Function

You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an alternative browser.

Discontinuous function $U$ with continuous preferences can be written as a composition of discontinuous & monotone function and a continuous function

• Thread starter dictatemetokcus
• Start date Apr 7, 2022

• #1

dictatemetokcus Asks: Discontinuous function $U$ with continuous preferences can be written as a composition of discontinuous & monotone function and a continuous function
Let $U : X \subseteq \mathbb{R}^n \to \mathbb{R}$; $n \geq 1$.

Conjecture: Every discontinuous utility function $U$ representing continuous preferences can be written as $U = f \circ g$ for some continuous $g$ and discontinuous strictly monotone $f$.

The goal is to prove or disprove this conjecture.

(Equivalent) Definitions of Continuous preferences:

• For any $y \in X$, the sets $\{x \in X : U(x) \geq y\}$ and $\{x \in X : y \geq U(x)\}$ are closed.
• If $U(x) > U$, then $\exists$ open balls $B_x$ and $B_y$ around $x,y$ (respectively) such that $(r \in B_x) \land (s \in B_y) \implies U(r) > U(s)$.

Treat the Utility function as a regular function.

SolveForum.com may not be responsible for the answers or solutions given to any question asked by the users. All Answers or responses are user generated answers and we do not have proof of its validity or correctness. Please vote for the answer that helped you in order to help others find out which is the most helpful answer. Questions labeled as solved may be solved or may not be solved depending on the type of question and the date posted for some posts may be scheduled to be deleted periodically. Do not hesitate to share your response here to help other visitors like you. Thank you, solveforum.

• HAMOODY 001
• 4 minutes ago
• Language & Culture
• Replies: 0

HAMOODY 001 Asks: What is the meaning of 一端 here?

そのレポートには　報道されていない東京卍會の行状の一端が記されています

in that report, you'll find some of what the Tokyo Manji Gang has been up to that hasn't been reported by the media.

is it a counter or has a different meaning?

SolveForum.com may not be responsible for the answers or solutions given to any question asked by the users. All Answers or responses are user generated answers and we do not have proof of its validity or correctness. Please vote for the answer that helped you in order to help others find out which is the most helpful answer. Questions labeled as solved may be solved or may not be solved depending on the type of question and the date posted for some posts may be scheduled to be deleted periodically. Do not hesitate to share your thoughts here to help others.

• NielIGuess
• 4 minutes ago
• Social
• Replies: 0

NielIGuess Asks: What should I do when a player who plays clerics doesn't want to use any of my deities?
So, my D&D group is pretty set in stone and has been for a while. One player almost always plays Clerics. Its his favorite class. I'm going to be the DM for a new campaign, this is my second time DMing and my first time DMing a homebrew campaign.
However, there's one problem: the type of cleric he wants to be doesn't exist in my setting.
There is no deity who matches up to the type of doctrine/ domain he wants. (He wants to play a type of nature worshipping cleric, but my setting's nature god doesn't work for multiple reasons).
I've asked him to play a different type of cleric/ a different class, but he doesn't want to. I don't want to have to add gods or change up my lore. What do I do?

SolveForum.com may not be responsible for the answers or solutions given to any question asked by the users. All Answers or responses are user generated answers and we do not have proof of its validity or correctness. Please vote for the answer that helped you in order to help others find out which is the most helpful answer. Questions labeled as solved may be solved or may not be solved depending on the type of question and the date posted for some posts may be scheduled to be deleted periodically. Do not hesitate to share your thoughts here to help others.

• Andrey
• 4 minutes ago
• Social
• Replies: 0

Andrey Asks: Can I stay in Romania for more than 90 days in a 180-day period by combining a short-term C visa and a German residence permit?
I am a non-EU citizen, a German resident with a long-term EC residence permit (Daueraufenthalt-EU).

With my German residence permit, I can stay in Romania for 90 days in any 180-day period, as Romania recognizes it as equivalent to a national visa (Decision 565/2014/EU). I have almost maxed out my 90 day stay and would like to be able to return to Romania without waiting 90 days for the current period to reset.

I had a conversation with the border police, and they suggested that I get a short stay Romanian national visa (type C, private visit). Thus, according to them:

1. I can stay 90 days in Romania on a national visa
2. Leave Romania
3. Re-enter using my German residence permit

This way the total length of stay may exceed 90 days in a 180-day period, but the stay on a national visa will not be considered in the calculation of days to stay on a residence permit and vice versa.

I was skeptical, but the border police officers insisted on the possibility of a combination of these stays. Moreover, on the website of the Ministry of Foreign Affairs, in the section on stays, there is the following sentence:

"Periods of stay authorised under a residence permit or a national long-stay visa are not taken into account when calculating the period of stay."

This, to be honest, may refer only to the Romanian residence permit.

My question: Сan I really combine periods of stay on a national Romanian short-term C visa and a German residence permit, potentially staying in Romania for up to 180 days in a 180-day period? Otherwise, can you please point me to the legal act that prohibits such a stay.

SolveForum.com may not be responsible for the answers or solutions given to any question asked by the users. All Answers or responses are user generated answers and we do not have proof of its validity or correctness. Please vote for the answer that helped you in order to help others find out which is the most helpful answer. Questions labeled as solved may be solved or may not be solved depending on the type of question and the date posted for some posts may be scheduled to be deleted periodically. Do not hesitate to share your thoughts here to help others.

• Zina
• 4 minutes ago
• Social
• Replies: 0

Zina Asks: Permesso di Soggiorno and Moldova citizenship
Can I travel to Mexico (tourism purposes) with a Permesso di Soggiorno and a biometric passport of Rep. of Moldova citizenship without visa?

SolveForum.com may not be responsible for the answers or solutions given to any question asked by the users. All Answers or responses are user generated answers and we do not have proof of its validity or correctness. Please vote for the answer that helped you in order to help others find out which is the most helpful answer. Questions labeled as solved may be solved or may not be solved depending on the type of question and the date posted for some posts may be scheduled to be deleted periodically. Do not hesitate to share your thoughts here to help others.

kulikowskidins1974.blogspot.com

Source: https://solveforum.com/forums/threads/discontinuous-function-u-with-continuous-preferences-can-be-written-as-a-composition-of-discontinuous-monotone-function-and-a-continuous-function.789374/

Share this post