I am following a paper proof which starts with the following constraint:
$$\forall v \in [a,b], \forall \tilde{v} \in [a,b]$$ $$f_{i}(v) \geq f_{i}(\tilde{v}) + (v-\tilde{v})g_{i}(\tilde{v})$$
In the proof, the author writes let $h = \tilde{v} - v$ therefore we have:
$$ f_{i}(v) \geq f_{i}(h+v) + h*g_{i}(h+v)$$
I understand that this is when $\forall v \in [a,b]$ However, what interval will $h$ be defined in? Would it be that all $\forall h \in [a,b]$?
If $v=a$ and $\tilde{v} = a$ then $h \not\in [a,b]$?
You can show that $h$ will verify $$-|a-b| \leq h \leq |a-b|$$
So indeed, often $h\not\in [a,b]$