Claim: Let $V: [0, T] (\subset \mathbb{R}) \to \mathbb{R}$ be a Lipschitz continuous function, i.e. $$|V(x,y)|:=|V(x)-V(y)| \leq K_1 | x - y |$$ for all $x,y$ in $[0, T] $, where $V(.,.)$ denotes the difference between values of $V$ and $K$ is a constant. Then, it follows that the difference can be bounded, i.e there exists a $\tilde{K}$ such that
$$ \sup_{x,y,x’,y’ \in [0,T] | x>y, x’>y’}| V(x,y) - V(x’,y’)| \\ \leq \tilde{K} \sup_{x,y,x’,y’ \in [0,T] | x>y, x’>y’} | ( x - y) - (x’- y’) | \\ = \tilde{K} T $$
I know there is a chance this may be false from academic vox populi, can you find a counterexample?
This post here is a follow-up to this question, so more background can be found there. I believe this specific case is more tractable: https://math.stackexchange.com/questions/4266346/how-to-prove-lipschitz-functions-are-not-lipschitz-in-the-p-variation-metric