Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function with
$\lim_{0\to +\infty}f(x)=\lim_{0\to -\infty}f(x)=0$
I need to show, that the function $f$ attains a minimum and/or a maximum.
To apply the extreme value theorem, the function must be defined on a closed and bounded interval $[a,b]$. So I think I can't use it here.
It's clear however, since both limits are zero and $f$ is continuous, that there must exist an upper bound $a\geq f(x),\forall x\in\mathbb{R}$ and a lower bound $b\leq f(x),\forall x\in\mathbb{R}$
But how can I prove it rigorously?
Best Answer
Without loss of generality, assume that $f(0)$ is positive. There is an $M$ such that $$|x| > M \implies |f(x)| < f(0)$$ given the limits at $\pm \infty$. Now apply your theorem to the interval $[-M, M]$ to conclude that $f$ not only attains a maximum, but does so within $[-M, M]$.
Do something similar for the minimum.