Let $A$ be a subset of $\Bbb{R}^2$ with the property that every continuous function $f:A\to\Bbb{R}$ has a maximum in $A$. I have to prove that $A$ is compact.
Is $A$ were a subset of $\Bbb{R}$ then we could take a function that is monotone over the components of $A$ and show that each of the components is closed and bounded hence proving $A$ to be compact. But couldn't generalize this idea to a subset of $\Bbb{R}^2$ that easily.
If $A$ is not compact, it is either not bounded or not closed. If it is not bounded, simply put $f(x) = |x|$, which is then unbounded on $A$. If $A$ is not closed, there is a limit point $y$ of $A$ which is not contained in $A$. Define $f(x) = \frac{1}{|x - y|}$. This is continuous on $A \subset \mathbb{R}^2 \setminus \{y\}$.