Assuming $f: \mathbb{R}^n \rightarrow \mathbb{R}$, and $F$ is a monotone transform of $f$. That is $f_t = F(f): \mathbb{R}^n \rightarrow \mathbb{R}$.
A monotone transform means that $\forall x_1,x_2\in\mathbb{R}^n, f(x_1)<f(x_2) \iff F(f(x_1))<F(f(x_2))$. That is, the ordinality of points on the domain of a function with respect to its image is preserved.
For example, let $f(x) = x$, and $F(f) = x^2 f$, then $f_t = x^2 x=x^3$. Or let $f(x)=sin(x)$, and $F(f) = 3 * f$, then $f_t = 3 sin(x)$. Both of these $F$ are monotone transforms of $f$. However, if $f(x)=sin(x)$ and $F(f)=x^2 f$, then $f_t=x^2 sin(x)$, and $F$ is not a monotone transform.
What can be said about the set of all monotone transforms $F$? That is, can monotonicity be expressed in some mathematical conditions, that if satisfied, can guarantee that a transformation $F$ of a function $f$ will be monotone?
For instance, if there are no turning points in $f$, then $\texttt{sign}\nabla_{x}f = \texttt{sign}\nabla_{x}F(f)\;\forall x$ is a sufficient condition to say that $F(f)$ is a monotone transformation of $f$. The actual gradient does not need to be equal, only that it is increasing or decreasing at the same time.
Is there a more generic set of conditions that can be solved to tell if $F$ will monotonically transform $f$ over some domain?