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If $f(x)$ is a continuous function on all of $\mathbb R$, with the property that $\sup_{x\in\mathbb R}|f(x)|\leq 1$. If this is the case, how I can test if the sup is attained or not? (i.e., if there exists at least $x_{o}\in \mathbb R$ such that $|f(x_{o})|\geq |f(x)|, \forall x\in \mathbb R$).

Should we have something like $\lim_{x\to\pm\infty}|f(x)|=0$, or something else?

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$\lim\sup_{x\to\pm\infty} |f(x)|<1$ is sufficient but not necessary. –  Will Sawin Jun 15 '12 at 17:39
Will is incorrect. It is neither. But MathOverflow is probably not the place for this question. –  Gerald Edgar Jun 15 '12 at 18:06
(he probably meant $\limsup |fx| < \sup |f(x)|$) @Catherine: a good place for your question is e.g. en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics –  Pietro Majer Jun 15 '12 at 18:22
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closed as too localized by Michael Renardy, unknown (google), Gerald Edgar, Pietro Majer, Felipe Voloch Jun 15 '12 at 18:25

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