Let $f$ be a nonzero continuous function on $\mathbb R$. If we write $\sup_{x\in \mathbb R}|f(x)|=c$, does it mean that there is (at least) one $x_{o}\in \mathbb R$ such that $|f(x_{o})|=c$? Just to be sure!
Edit: $|f(x_{o})|=c$ instead of $f(x_{o})=c$, my false:)