I am reading this paper for a presentation and am trying to understand lemma 2.7 on page 9 to 10. Specifically, I understand the proof up till the last paragraph on page 9 and the top paragraph on page 10.
What does it mean the paper says "Since E and the Cantor Set are totally disconnected, they are nowhere dense and thus at most one continuation $F:[m, M] \rightarrow [0.1]$ ? I think the last part is saying that there exists at most one function that maps $F:[m, M] \rightarrow [0.1]$ that ensures a homeomorphism between the two spaces but what does it mean to be "nowhere dense" and what does it have to do with it?
How does the author then arrive at the fact that $F(x) = sup(F(y) : y \notin E, y\leq x)$? I would write out what I do know about this but I have no idea as to how this conclusion is arrived at.
On page 10, what is the contradiction that shows that the inverse function $g(x)$ is continuous. If I accept the $F(x)$ function and its properties from question 2 at face value then I understand that $f: E \rightarrow C$ is continuous and monotone increasing. But how exactly is $g: f^{-1}$ also continuous?