Let $f$ and $g$ be two functions defined on some subset $S$ of $R^n.$ Is it true that

$ma{x}_{\text{x}\text{i}\text{n}\text{S}}\{f(x)+g(x)\}=ma{x}_{\text{x}\text{i}\text{n}\text{S}}f(x)+ma{x}_{\text{x}\text{i}\text{n}\text{S}}g(x)?$

$($If it's true, give a proof. If not, give a counter example.$)$