User catherine - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:40:23Z http://mathoverflow.net/feeds/user/24487 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99728/supremum-of-continuous-functions-on-mathbbr Supremum of continuous functions on $\mathbb{R}$ Catherine 2012-06-15T17:36:16Z 2012-06-15T17:36:16Z <p>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 $|f(x_{o})|\geq |f(x)|, \forall x\in \mathbb R$).</p> <p>Should we have something like $\lim_{x\to\pm\infty}|f(x)|=0$, or something else?</p>