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Given a continuous and increasing function $f$ on $\mathbb R$, and a given point $x_{0}\in \mathbb R$, what we can say about $$ f(x+x_{0})-f(x)$$ for all $x\in \mathbb R$? Do we have a bound for this difference? I forget to say that $f'$ is bounded on $\mathbb R$. Does this change anything!

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    Take $f(x)=e^x$, so $f(x+x_0)-f(x) = e^x(e^{x_0}-1)$ which is unbounded. There isn't a whole lot you can say about this difference in general - what sort of thing are you looking to do?2011-04-14
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    @Alon: Note the later comment "I forget to say that $f'$ is bounded". I.e., assume $0\leq f'(x)\lt M$ for some fixed $M$.2011-04-14
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    @Arturo: well that does change things quite a bit...2011-04-14

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