Possibility of a number line that has variable density. In my real analysis class, I have been informed that if you have a continuous line say $[0,1]$, and you do a mapping say $A\to B$ such that any element in $A$ is equal to $B^2$, where $A$ is every element in $[0,1]$  you would ultimately get a line that is denser toward the $0$ and less dense toward the $1$.
I have two question regarding this result,


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*Can we note that every element in $A$, say variable $a$  has both its square and square root already existing in the line $[0, 1]$, therefore the mapping is only an exchange of the position of points, 


for example we know that if we have point $0.5$  
both $0.5^2$ and $\sqrt{0.5}$ exist in $[0,1]$  therefore if we do the mapping, $0.5 \mapsto 0.25$ doesn't the other point $\sqrt{0.5}$ replaces $0.5$'s original position ?


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*How is it possible that you can have a denser part in a continuous line, does this not ultimately suggest that there will be a gap around the $1$'s side?


Any comment would be much appreciated
 A: All that is meant by the statement that the line is "more dense" towards 0 is this: say that you pick some small number $\epsilon>0$. Then the interval $[0,\sqrt{\epsilon}]$ all gets mapped (by $x\mapsto x^2$) in to the interval $[0,\epsilon]$; so, the preimage of $[0,\epsilon]$ under this map has total length $\sqrt{\epsilon}$.
On the other side, the numbers $[\sqrt{1-\epsilon},1]$ are mapped by $x\mapsto x^2$ to the interval $[1-\epsilon,1]$; so, in this case, the preimage of $[1-\epsilon,1]$ has length $1-\sqrt{1-\epsilon}$.
The interesting thing here: even though the intervals $[0,\epsilon]$ and $[1-\epsilon,1]$ have the same length... their preimages do not.  In fact, for every $\epsilon\in(0,1)$, it turns out that $\sqrt{\epsilon}$ (the "size" of the preimage of $[0,\epsilon]$) is larger than $1-\sqrt{1-\epsilon}$ (the "size" of the preimage of $[1-\epsilon,1]$).
So, all that they really meant was that different intervals of $[0,1]$ get more or less "spread out" by the mapping $x\mapsto x^2$.
