I was reading the an article which made the following statement:
since $g$ is a continuous function which is bounded from below by a positive number, it has a continuous inverse?
The only other relevant information about $g$ is that it is a function from a compact set to $\mathbb{R}$, but that is used in the statement that is attains its minimum. My question is:
Why exactly can one conclude $g$ has a continuous inverse?