Min/max of functions restricted to subsets of their domains

[V0ODO0CH1LD]

If the critical points corresponding to the global min/max of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ lie in a subset $A$ of $\mathbb{R}^2$, then the global min/max of $f$ in $A$ correspond to the global min/max of $f$.

If the global min/max of $f$ lie outside of $A$, then either the global min/max of $f$ in $A$ are in the boarders of $A$ or are the local min/max of $f$ that lie in $A$. (There is no way that the global min/max of $f$ in $A$ is not a critical point of $f$ or in the boarders of $A$, right?)

And finally if the global min/max of $f$ in $A$ lie in the boarders of $A$, then either the subset $A$ is closed in which case the global min/max corresponds to those points in the boarders, or the subset $A$ is open in which case (say that the point in the boarder corresponding to the max is the point $(a,b)$) the global max would then be:
\lim_{(x,y)\rightarrow{}(a,b)}f(x,y).
Is that right? Did I miss something? Are there cases I left out?

By the way, I think that what I mean when I say the global min max of $f:\mathbb{R}^2\rightarrow\mathbb{R}$ in $A$ is the global min/max of the restricted function $f:A\rightarrow\mathbb{R}$ although I am not entirely sure.

 

[HallsofIvy]

If has a global max or min then it must occur where the derivative is 0 or does not exist. So if you are requiring that all "critical points" occur in A, then, yes, any global max or min must be in A. Of course, the function may not have a global max, f(x) steadily increasing as x increases, for example.