When do we know that $\sup_{x \in \mathbb{R}} f(x) \in f(\mathbb{R})$? And is there a argsup function or something equivalent?

Question

Suppose we have a function $f\colon\mathbb{R}\to A$ where $A \subseteq \mathbb{R}$ is bounded from above. Then we know $\sup_{x \in \mathbb{R}} f(x)$ exists. My questions are:

• Which other assumptions do we need to know that $\sup_{x \in \mathbb{R}} f(x) \in f(\mathbb{R})$
• And how can we denote the element $x \in \mathbb{R}$ which corresponds to the supremum. Is there some sort of argsup function?