I'm trying to define the collection of functions $H:V\to\mathbb{R}$ s.t. $\forall h\in H\exists f,g\in V\to \mathbb{R}$ s.t. $\forall v\in V$:
$f$ "makes" $h$ descend.
$g$ "makes" $h$ ascend.
For example, supposing $\forall v\in V:f(v)> 1,g(v)> 1$: $$h(v)=f(v)-g(v)$$ $$h(v)=\frac{f(v)}{g(v)}$$ $$h(v)=\sqrt[g(v)]{f(v)}$$
The definition I came up with is: $$h\in \{ H\in 2^{G:V\to \mathbb{R} }\to \mathbb{R} \| \exists \{ f,g\} \subseteq V\to \mathbb{R} , \forall v\in V\to \mathbb{R} : \frac{\partial}{\partial f}H(v)>0 , \frac{\partial}{\partial g}H(v)<0\}$$ I feel like something is off.
Can you please help me out?