What does it mean that $f$ achieves its minimum

Question

Let $f:C\to \mathbb{R}$ be a function such that $x_0\in\mathbb{R}$ exists such that
$$f(x_0) =\inf_{x\in C} f(x).$$

Does it mean that $f$ achieves its minimum?

It is not clear for me, I hope someone could help. Thank you in advance!

Answer 1

Yes, this is what "achieves its minimum" means. To put it another way, a function $f$ achieves its minimum on a set $X$ if there is an $x_0\in X$ such that $f(x_0)\le f(x)$ for all $x\in X$.

There is an important theorem relating to this: if $f$ is continuous on a nonempty closed interval, then it achieves its minimum and maximum values on that interval.