what does "pasting" and "cut" mean in function (topology)

Question

There is an example that explain this, but I still have 3 question:

Let $f: [0, 1] ∪] 2, 3] → [0, 2]$ the application given by

$f (x) = x$ if $0 ≤ x ≤ 1$

$f(x)=x - 1$ if $2 <x ≤3$

Despite being bijective, $f$ is "pasting" the intervals $[0, 1]$ and$] 2, 3]$ in the interval $[0, 2]$, so $f^{−1}$ "Cut" this one at point $1$.

In general, if a continuous application is an application that does not cut, although can paste, a homeomorphism is an application that does not cut or paste. For that it does not paste has to be bijective, but we have just seen that this is not enough.

1. What does "pasting" and "cut" mean?
2. Why continuous function is an application that does not cut?
3. Why homeomorphism is an application that does not cut or paste?