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Suppose that $f: \mathbb R \to \mathbb R$ is increasing for $x > 0$ such that $f(x) < f(\infty)$. What doest the last inequality mean? Does that mean $f(x)$ is bounded? As an example, let $f(x)=x$, does that mean $f(x) < f(\infty)$?

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    $\begingroup$ In which context are you encountering this? This is, at best, non-standard notation. $\endgroup$
    – Thorgott
    Commented Apr 13, 2020 at 23:40

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The way I'd read it is that $f(\infty)$ stands for the extended real number $\lim_{x\to \infty}f(x)$ (i.e. possibly $f(\infty)=\infty$) and that the condition is supposed to be $f(x)<f(\infty)$ for all $x\in\Bbb R$. In the case $f(\infty)=\infty$ the condition should add nothing. In the case $f(\infty)\in\Bbb R$, the condition should add the fact that $f(x)<f(\infty)$ for all $x\le 0$, and that $f$ is not eventually constant as $x\to\infty$.

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