I have a fixed function [itex]U[/itex] and a function [itex]f[/itex](adsbygoogle = window.adsbygoogle || []).push({});

that I want to know something about, both from

[itex]\mathbb{R}^{n}\rightarrow\mathbb{R}[/itex], for which I know

condition (G) that [itex]U(x)\geq{}U(y)[/itex] implies

[itex]f(x)\geq{}f(y)[/itex]. [itex]f=U[/itex] and [itex]f=x_{0}[/itex]

for any [itex]x_{0}\in{}\mathbb{R}[/itex] trivially fulfill (G). Which

[itex]f[/itex] fulfills (G) non-trivially? Are there additional

assumptions (continuity for example) which would allow me to claim

that there is no [itex]f[/itex] that non-trivially fulfills (G)?

This question will help me solve a set-theoretic puzzle based on

Debreu's theorem -- that's why I've put the thread in this category.

[itex]U[/itex] stands for the utility function, [itex]f[/itex] for a less discriminating

utility function that is not trivially [itex]f=U[/itex] or constant.

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# Two real-valued functions on a real vector space

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