how to ensure existence and uniqueness of the maximizer of a function with not compact domain

Question

Let $f: A \to \mathbb{R}$ be a function and $A\subset \mathbb{R}^d$. It looks like I can ensure existence and uniqueness of a maximizer of $f$ if I let $f$ be strictly concave and upper semicontinuous with bounded super-level sets. I wonder why this is true and moreover what conditions should I place on $A$. I would like to avoid $A$ being bounded. I wonder if I can let $A$ be an open or closed rectangle.