The Boundedness Theorem and the Extreme Value Theorem for subsets of finite-dimensional (Euclidean) space
In an earlier post we looked at the Boundedness Theorem and the Extreme Value Theorem for continuous, real-valued functions on closed and bounded intervals $[a,b]$. However, the proof we gave there generalises to any setting in which the lemma we used is valid. Let $D$ be a non-empty subset of $\R^2$, and consider the following property (B) ( non-standard! ) that $D$ may or may not have: (B) Every continuous function $f:D\to\R$ is bounded above on $D$. By the reasoning in the earlier post , we can show that the Extreme Value Theorem holds for any non-empty set $D$ that satisfies condition (B). So the question arises, which subsets of $\R^2$ satisfy condition (B)? By considering the continuous function $f(x,y)=\sqrt{x^2+y^2}$ (or we could use $f(x,y)=x^2+y^2$), it is clear that any subset $D$ of $\R^2$ that satisfies condition (B) must be a bounded subset of $\R^2$ (that is, the set of distances from points of $D$ to $(0,0)$ is bounde...