I am interested in two-variable functions f with the following property:

$\forall M, \exists N$ such that $\forall x, \forall y>N, f(x,y)>M.$

(To be absolutely clear: the $\forall x$ quantifier is not restricted like the $\forall y$ one is)

Is there a standard name for such functions?

One particular example (on $\mathbb{N}$): if $\lim_{y\to\infty}h(y)=\infty$, then $f(x,y):=\sum_{i=\min(x,y)}^yh(i)$ has the above property.