Here is a qualifying exam question which I hope someone can help me with. I have done all of it except having problem to "visualize" $\partial S$ and hence have no idea, is the intersection of $K \cap \partial S$ empty or not?
Here is the question
Assume $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a continuous function such that
(a) there exist points $x_0$ and $x_1\in \mathbb{R}^n$ with $f(x_0)=0$ and $f(x_1)=3$,
(b) there exist positive constants $C_1$ and $C_2$ such that $f(x)\geq C_1|x|-C_2$ for all $x\in \mathbb{R}^n$.
Let $S:=\{ x\in\mathbb{R}^n : f(x)<2 \}$ and let $K:=\{ x\in\mathbb{R}^n : f(x)\leq 1 \}$. Define the distance from $K$ to $\partial S$ (the boundary of $S$) by the formula
$ \text{dist} (K, \partial S) := \inf_{p\in K, q\in \partial S} |p-q|. $
Prove that the dist$(K,\partial S) >0$. Then give an example of a continuous function $f$ satisfying (a) but dist$(K,\partial S) =0$.
End of question.
As you can see $K=f^{-1}((-\infty, 1])$ and $K\subset \{ x\in \mathbb{R}^n : ||x||\leq \frac{1+C_2}{C_1} \}$ is closed and bounded, surely $\partial S$ is closed, hence I just needed the empty intersection to draw the final conclusion.
Also is there any easy counterexample that needed to satisfied (a)?
Thanks