Two real-valued functions on a real vector space

[noowutah]

I have a fixed function U and a function f
that I want to know something about, both from
\mathbb{R}^{n}\rightarrow\mathbb{R}, for which I know
condition (G) that U(x)\geq{}U(y) implies
f(x)\geq{}f(y). f=U and f=x_{0}
for any x_{0}\in{}\mathbb{R} trivially fulfill (G). Which
f fulfills (G) non-trivially? Are there additional
assumptions (continuity for example) which would allow me to claim
that there is no f 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.
U stands for the utility function, f for a less discriminating
utility function that is not trivially f=U or constant.

 

[Stephen Tashi]

Which f fulfills (G) non-trivially?

I think we need a definition of "non-trivially". Is f(x) = 3 U(x) + 6 a trivial or non-trivial fullfillment?

 

[noowutah]

Sorry, yes, I left out something important. I need f to
be LESS discriminating than U. So, I want
\exists{}x,y\in\mathbb{R}^{n} with x\neq{}y
and f(x)=f(y), whereas U(x)>U(y).

I think I've answered my own question. Yes, there is a continuous
f (given a continuous U) which fulfills (G)
is neither U nor constant. The only constraint is that,
if the functions are differentiable, f'(x)U'(x) is
non-negative. For my less-discriminating condition, the function must
be zero somewhere where U is monotonically increasing.

Thanks, I think I can take it from here.