Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$.
(I'm not specific about the domain of definition, I have in mind functions from $\mathbf{R}_+$ to itself but I don't expect this to be important, it could be from any interval to any other, or even from any totally ordered set to itself.)
If I had to coin a terminology I would say something like "dynamically non-decreasing" but it's a bit ugly.