Prove if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function and $\lim_{x\to\pm\infty} f(x) = 0$, $f$ has a global maximum and minimum.

Question

Prove if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function and $\lim_{x\to\pm\infty} f(x) = 0$, then $f$ has a global maximum and minimum.

This is the exact question posed, but wouldn't a function such as the bell curve be a counterexample as it would only have a global maximum?

Answer 1

A continuous function does not blow up anywhere else except at infinity. Since the limit is $0$ at infinity, one can conclude that $f(\mathbb{R})=]a,b[$.