Prove that f has at least one global minimizer

Question

$f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function such that

$\displaystyle\lim_{\|x\| \to \infty} f(x) = \infty$

On a side note: how can a function have more than one global minimizer? Is a global minimizer not, by definition, $f(x^*) \le f(x)$ for all $x$?

Answer 1

The conditions you've presented are not enough to prove that $f$ is convex.

Hint for proving that $f$ has an absolute minimum:

By definition of the limit, there exists a $\rho > 0$ such that $f(x) > f(0)$ whenever $\|x\| > \rho$.

Note that $f$ must attain a minimum on $\{x:\|x\| \leq \rho\}$. Why is this also the absolute minimum?