assuming we have two smooth function ${f\_1},{f\_2}:{R^N} \to R$, 


1. under what condition, we have


${f\_1}\left( {{{\bf{x}}\_1}} \right) \ge {f\_1}\left( {{{\bf{x}}\_2}} \right) \leftrightarrow {f\_2}\left( {{{\bf{x}}\_1}} \right) \ge {f\_2}\left( {{{\bf{x}}\_2}} \right), \forall \bf{x}\_1\neq \bf{x}\_2\in{R^N}$ 


2. what property is called in mathematical terminology about $f\_1,f\_2$?


we can assume both of the function are convex if needed.
I know that monotone transformation preserve the above property. but I am not sure if it is necessary to be able to define a monotone transformation between these two functions or not?



monotonicity alike functions
Let Property A be the property described above.


Let Property B be the property that $f\_2=g \circ f\_1$ for some strictly monotonically increasing function $g:R\rightarrow R$.


Then Property A and Property B are equivalent.


To see that Property B implies Property A, suppose that $f\_1(x\_1)\geq f\_1(x\_2)$. Since $g$ is monotonic, if we apply $g$ we get $f\_2(x\_1)\geq f\_2(x\_2)$. Since $g$ strictly monotonic implies that $g^{-1}$ is also strictly monotonic, we get the other direction.


To see that Property A implies Property B, assume that Property A holds. Then $f\_1(x\_1)=f\_1(x\_2)$ iff $f\_2(x\_1)=f\_2(x\_2)$. Therefore, we can define $g$ on $Im(f\_1)$ by setting $g(f\_1(x))=f\_2(x)$; by the preceding observation, $g$ is well-defined, and $f\_2=g\circ f\_1$.


