As we know that: If $f$ and $g$ are bounded functions on some arbitrary set, then $ |\sup f - \sup g | \leq \sup |f-g| $.
Indeed, $\sup f = \sup (f -g + g) \leq \sup (f-g) + \sup g \leq \sup |f-g| + \sup g$. Then, it implies $\sup f - \sup g \leq \sup |f-g|$. By changing the roles of $f$ with $g$, we finally obtain $ |\sup f - \sup g | \leq \sup |f-g| $.
However, I am wondering that if it is even possible to extend $f$ and $g$ be any unbounded functions and above property still holds?
Thanks so much in advance!