Prove for each function $f:\mathbb Q\times\mathbb Q\to\mathbb R$ there exists a function $g:\mathbb Q\to\mathbb R$ such that $f(x,y)\le g(x)+g(y)\,\forall x,y\in\mathbb Q$.

Find a function $f:\mathbb R\times\mathbb R\to\mathbb R$ for which there is no function $g:\mathbb R\to\mathbb R$ such that $f (x,y)\le g (x)+g (y)\,\forall x,y\in\mathbb R $.