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Given a function $f(x)$, we can define the new function $$ A_f(t) = \limsup\limits_{x\to\infty}\ (f(x+t) - f(x)) $$

Is there a place that this transformation has been studied?

Also, given a positive real number $r$, I'm interested in the space of functions such that $A_f(t)$ exists for all $t>0$ and grows at most a linear rate less than $r$. More precisely, there exists an $a<r$ and an $s$ such that for all $t>s$ $$ A_f(t) - a t \le 0 $$

Is there a nice characterization of this space or perhaps some good sized subset of it? It's easy to see, for example, that it includes the log of any polynomial and linear functions with coefficient less than r.

(This question actually arose studying the exponentiated version of the above, ie, $$ \limsup\limits_{t->\infty}\frac{g(x+t)}{g(x)} $$ which might make the latter condition seem a little less strange.)

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well, each function that has bounded derivative at infinity satisfies this condition (such as the log of polynomials etc.) –  leshik May 23 '13 at 3:33
    
That does seem to be sufficient. Not sure why I missed that. Thanx. –  user60338 May 29 '13 at 18:50

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