Is set Dedekind continuous? The set is called Dedekind continuous when there is no way to make a $A||A'$ or $A)(A'$ cuts where:


*

*$A||A'$ cut means there is the biggest element in $A$ set and the lowest element in $A'$ set

*$A)(A'$ cut means there is no biggest element in $A$ and no lowest element in $A'$
We all know that $\mathbb{R}$ is Dedekind continuous.
The question is whether the following subset of $\mathbb{R}$ is Dedekind continuous or not:
$$ M = (-\infty; 2) \cup [3; +\infty) $$
It is obvious this set can't be "geometry" continuous since there is a gap between $2$ and $3$ but I can't find a way to make $A||A'$ or $A)(A'$ cuts to prove it is not Dedekind continuous.
 A: This set is Dedekind continuous. In fact, it is order isomorphic to the real line through the following map:
$$
f: M\to \Bbb R\\
f(x) = \cases{x & if $x<2$\\
x-1 & if $x \geq 3$}
$$
A: This set is Dedekind continuous.

It is obvious this set can't be "geometry" continuous since there is a gap between 2 and 3...

At first glance, the fact this set is Dedekind continuous actually sounds weird. There is "gap" between $2$ and $3$. How this could be considered continuous in any way!?

But this is not what actually happens. The set $ [3; +\infty)$ comes right after the $(-\infty; 2)$ so there is no gap between them.  The true image would be:

As you can clearly see, moving from the left to right we gradually and continously move from elements infinitely close to $3$ right to element $3$ without any "jumping the gaps".
Dedekinds cuts can be applied to any totally ordered sets (including sets of numbers). So the imaginary "gape" is just a kind of a mind trick which occurs because we think of these sets as bunches of numbers that represent a cetrain value rather then thinking of them as sets of impersonal ordered elements or even dots.
If we name $3$ as "element $b$" the pharse "gradually move from elements that lie to left of $b$ to element $b$ itself" does not sound weird at all.
Heading back to "geometric continuity". Imagine we cut a $[\ldots)$ section of a straight line. We can easily glue it, so it won't lose its "geometric continuity".

The following sets are Dedekind continuous as well:


*

*$ (-\infty; 1) \cup [5; +\infty) $

*$ [2; 11] \cup (54; 108) $

*$\cdots$

It is also worth mentioning that the following subsets of $\mathbb{R}$ are not Dedekind continuous:


*

*$ (-\infty; 2] \cup [3; +\infty) $ since we can make $A||A'$ cut

*$ (-\infty; 1) \cup (5; +\infty) $ since we can make $A)(A'$ cut

*$\ldots$
