Let $A$ be a subset of $\mathbb { R } ^ { 2 }$ with the property that every continuous function $f : A \rightarrow \mathbb { R }$ has a maximum in $A$. Prove that $A$ is compact.
Let $A$ be a subset of $\mathbb { R } ^ { 2 }$ with the property that every continuous function $f : A \rightarrow \mathbb { R }$ has a maximum in $A$. Prove that $A$ is compact.