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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.

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    $\begingroup$ A function with $f(x)<x$ is sometimes called regressive. $\endgroup$ – Achilles Apr 18 '15 at 14:26
  • $\begingroup$ I am not aware of any terminology for exactly those functions, but you might want to look into Landau Notation which can be used for expressing differend bounds of functions. $\endgroup$ – flawr Apr 18 '15 at 14:26
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    $\begingroup$ @flawr: I don't mean anything asymptotic. I really mean what I wrote and Landau notation is irrelevant $\endgroup$ – YCor Apr 18 '15 at 14:36
  • $\begingroup$ I don't think the downvote is fair. The condition ($\forall x,f(x)\ge x$) is very natural in dynamics, when we study how the sequence $(x,f(x),f(f(x)),\dots,f^{\circ n}(x),\dots)$ behaves (then it is an non-decreasing sequence). $\endgroup$ – YCor Apr 18 '15 at 15:43

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